KILLED proof of input_aBYdzIgL4I.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (6) CdtProblem (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxRelTRS (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxRelTRS (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (12) typed CpxTrs (13) OrderProof [LOWER BOUND(ID), 5 ms] (14) typed CpxTrs (15) RewriteLemmaProof [LOWER BOUND(ID), 331 ms] (16) BEST (17) proven lower bound (18) LowerBoundPropagationProof [FINISHED, 0 ms] (19) BOUNDS(n^1, INF) (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 109 ms] (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 171 ms] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 127 ms] (26) typed CpxTrs (27) RewriteLemmaProof [LOWER BOUND(ID), 174 ms] (28) typed CpxTrs (29) RewriteLemmaProof [LOWER BOUND(ID), 54 ms] (30) typed CpxTrs (31) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (32) CdtProblem (33) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CpxRelTRS (39) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (40) CpxTRS (41) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CpxWeightedTrs (43) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CpxTypedWeightedTrs (45) CompletionProof [UPPER BOUND(ID), 0 ms] (46) CpxTypedWeightedCompleteTrs (47) NarrowingProof [BOTH BOUNDS(ID, ID), 92 ms] (48) CpxTypedWeightedCompleteTrs (49) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CpxRNTS (53) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 89 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 4 ms] (60) CpxRNTS (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 119 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 72 ms] (66) CpxRNTS (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 449 ms] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 105 ms] (72) CpxRNTS (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (74) CpxRNTS (75) IntTrsBoundProof [UPPER BOUND(ID), 310 ms] (76) CpxRNTS (77) IntTrsBoundProof [UPPER BOUND(ID), 12 ms] (78) CpxRNTS (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (80) CpxRNTS (81) IntTrsBoundProof [UPPER BOUND(ID), 392 ms] (82) CpxRNTS (83) IntTrsBoundProof [UPPER BOUND(ID), 84 ms] (84) CpxRNTS (85) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (86) CpxRNTS (87) IntTrsBoundProof [UPPER BOUND(ID), 60 ms] (88) CpxRNTS (89) IntTrsBoundProof [UPPER BOUND(ID), 103 ms] (90) CpxRNTS (91) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (92) CpxRNTS (93) IntTrsBoundProof [UPPER BOUND(ID), 432 ms] (94) CpxRNTS (95) IntTrsBoundProof [UPPER BOUND(ID), 235 ms] (96) CpxRNTS (97) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (98) CpxRNTS (99) IntTrsBoundProof [UPPER BOUND(ID), 715 ms] (100) CpxRNTS (101) IntTrsBoundProof [UPPER BOUND(ID), 176 ms] (102) CpxRNTS (103) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (104) CpxRNTS (105) IntTrsBoundProof [UPPER BOUND(ID), 1227 ms] (106) CpxRNTS (107) IntTrsBoundProof [UPPER BOUND(ID), 518 ms] (108) CpxRNTS (109) CompletionProof [UPPER BOUND(ID), 0 ms] (110) CpxTypedWeightedCompleteTrs (111) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (112) CpxRNTS (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 5 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 2 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 6 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 22 ms] (170) CdtProblem (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 1 ms] (184) CdtProblem (185) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 12 ms] (190) CdtProblem (191) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtRewritingProof [BOTH BOUNDS(ID, ID), 23 ms] (200) CdtProblem (201) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 204 ms] (204) CdtProblem (205) CdtRewritingProof [BOTH BOUNDS(ID, ID), 13 ms] (206) CdtProblem (207) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (210) CdtProblem (211) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (212) CdtProblem (213) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (214) CpxWeightedTrs (215) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (216) CpxTypedWeightedTrs (217) CompletionProof [UPPER BOUND(ID), 0 ms] (218) CpxTypedWeightedCompleteTrs (219) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (220) CpxTypedWeightedCompleteTrs (221) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (222) CpxRNTS (223) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (224) CpxRNTS (225) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (226) CpxRNTS (227) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (228) CpxRNTS (229) IntTrsBoundProof [UPPER BOUND(ID), 206 ms] (230) CpxRNTS (231) IntTrsBoundProof [UPPER BOUND(ID), 77 ms] (232) CpxRNTS (233) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (234) CpxRNTS (235) IntTrsBoundProof [UPPER BOUND(ID), 487 ms] (236) CpxRNTS (237) IntTrsBoundProof [UPPER BOUND(ID), 157 ms] (238) CpxRNTS (239) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (240) CpxRNTS (241) IntTrsBoundProof [UPPER BOUND(ID), 293 ms] (242) CpxRNTS (243) IntTrsBoundProof [UPPER BOUND(ID), 66 ms] (244) CpxRNTS (245) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (246) CpxRNTS (247) IntTrsBoundProof [UPPER BOUND(ID), 434 ms] (248) CpxRNTS (249) IntTrsBoundProof [UPPER BOUND(ID), 126 ms] (250) CpxRNTS (251) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (252) CpxRNTS (253) IntTrsBoundProof [UPPER BOUND(ID), 704 ms] (254) CpxRNTS (255) IntTrsBoundProof [UPPER BOUND(ID), 408 ms] (256) CpxRNTS (257) CompletionProof [UPPER BOUND(ID), 0 ms] (258) CpxTypedWeightedCompleteTrs (259) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (260) CpxRNTS ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: min(x, 0) -> 0 min(0, y) -> 0 min(s(x), s(y)) -> s(min(x, y)) max(x, 0) -> x max(0, y) -> y max(s(x), s(y)) -> s(max(x, y)) minus(x, 0) -> x minus(s(x), s(y)) -> s(minus(x, any(y))) gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y))) any(s(x)) -> s(s(any(x))) any(x) -> x S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: min(x, 0') -> 0' min(0', y) -> 0' min(s(x), s(y)) -> s(min(x, y)) max(x, 0') -> x max(0', y) -> y max(s(x), s(y)) -> s(max(x, y)) minus(x, 0') -> x minus(s(x), s(y)) -> s(minus(x, any(y))) gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y))) any(s(x)) -> s(s(any(x))) any(x) -> x S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: min(x, 0) -> 0 min(0, y) -> 0 min(s(x), s(y)) -> s(min(x, y)) max(x, 0) -> x max(0, y) -> y max(s(x), s(y)) -> s(max(x, y)) minus(x, 0) -> x minus(s(x), s(y)) -> s(minus(x, any(y))) gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y))) any(s(x)) -> s(s(any(x))) any(x) -> x S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) gcd(s(z0), s(z1)) -> gcd(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))) any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 Tuples: MIN(z0, 0) -> c MIN(0, z0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(z0, 0) -> c3 MAX(0, z0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(z0, 0) -> c6 MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) ANY(z0) -> c12 S tuples: MIN(z0, 0) -> c MIN(0, z0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(z0, 0) -> c3 MAX(0, z0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(z0, 0) -> c6 MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) ANY(z0) -> c12 K tuples:none Defined Rule Symbols: min_2, max_2, minus_2, gcd_2, any_1 Defined Pair Symbols: MIN_2, MAX_2, MINUS_2, GCD_2, ANY_1 Compound Symbols: c, c1, c2_1, c3, c4, c5_1, c6, c7_2, c8_3, c9_3, c10_2, c11_1, c12 ---------------------------------------- (7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (8) 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: MIN(z0, 0) -> c MIN(0, z0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(z0, 0) -> c3 MAX(0, z0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(z0, 0) -> c6 MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) ANY(z0) -> c12 The (relative) TRS S consists of the following rules: min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) gcd(s(z0), s(z1)) -> gcd(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))) any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 Rewrite Strategy: INNERMOST ---------------------------------------- (9) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (10) 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: MIN(z0, 0') -> c MIN(0', z0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(z0, 0') -> c3 MAX(0', z0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(z0, 0') -> c6 MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) ANY(z0) -> c12 The (relative) TRS S consists of the following rules: min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) max(z0, 0') -> z0 max(0', z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) gcd(s(z0), s(z1)) -> gcd(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))) any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 Rewrite Strategy: INNERMOST ---------------------------------------- (11) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (12) Obligation: Innermost TRS: Rules: MIN(z0, 0') -> c MIN(0', z0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(z0, 0') -> c3 MAX(0', z0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(z0, 0') -> c6 MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) ANY(z0) -> c12 min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) max(z0, 0') -> z0 max(0', z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) gcd(s(z0), s(z1)) -> gcd(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))) any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 Types: MIN :: 0':s -> 0':s -> c:c1:c2 0' :: 0':s c :: c:c1:c2 c1 :: c:c1:c2 s :: 0':s -> 0':s c2 :: c:c1:c2 -> c:c1:c2 MAX :: 0':s -> 0':s -> c3:c4:c5 c3 :: c3:c4:c5 c4 :: c3:c4:c5 c5 :: c3:c4:c5 -> c3:c4:c5 MINUS :: 0':s -> 0':s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c11:c12 -> c6:c7 any :: 0':s -> 0':s ANY :: 0':s -> c11:c12 GCD :: 0':s -> 0':s -> c8:c9:c10 c8 :: c8:c9:c10 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10 minus :: 0':s -> 0':s -> 0':s max :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s c9 :: c8:c9:c10 -> c6:c7 -> c:c1:c2 -> c8:c9:c10 c10 :: c8:c9:c10 -> c:c1:c2 -> c8:c9:c10 c11 :: c11:c12 -> c11:c12 c12 :: c11:c12 gcd :: 0':s -> 0':s -> gcd hole_c:c1:c21_13 :: c:c1:c2 hole_0':s2_13 :: 0':s hole_c3:c4:c53_13 :: c3:c4:c5 hole_c6:c74_13 :: c6:c7 hole_c11:c125_13 :: c11:c12 hole_c8:c9:c106_13 :: c8:c9:c10 hole_gcd7_13 :: gcd gen_c:c1:c28_13 :: Nat -> c:c1:c2 gen_0':s9_13 :: Nat -> 0':s gen_c3:c4:c510_13 :: Nat -> c3:c4:c5 gen_c6:c711_13 :: Nat -> c6:c7 gen_c11:c1212_13 :: Nat -> c11:c12 gen_c8:c9:c1013_13 :: Nat -> c8:c9:c10 ---------------------------------------- (13) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: MIN, MAX, MINUS, any, ANY, GCD, minus, max, min, gcd They will be analysed ascendingly in the following order: MIN < GCD MAX < GCD any < MINUS ANY < MINUS MINUS < GCD any < minus minus < GCD max < GCD min < GCD minus < gcd max < gcd min < gcd ---------------------------------------- (14) Obligation: Innermost TRS: Rules: MIN(z0, 0') -> c MIN(0', z0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(z0, 0') -> c3 MAX(0', z0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(z0, 0') -> c6 MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) ANY(z0) -> c12 min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) max(z0, 0') -> z0 max(0', z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) gcd(s(z0), s(z1)) -> gcd(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))) any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 Types: MIN :: 0':s -> 0':s -> c:c1:c2 0' :: 0':s c :: c:c1:c2 c1 :: c:c1:c2 s :: 0':s -> 0':s c2 :: c:c1:c2 -> c:c1:c2 MAX :: 0':s -> 0':s -> c3:c4:c5 c3 :: c3:c4:c5 c4 :: c3:c4:c5 c5 :: c3:c4:c5 -> c3:c4:c5 MINUS :: 0':s -> 0':s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c11:c12 -> c6:c7 any :: 0':s -> 0':s ANY :: 0':s -> c11:c12 GCD :: 0':s -> 0':s -> c8:c9:c10 c8 :: c8:c9:c10 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10 minus :: 0':s -> 0':s -> 0':s max :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s c9 :: c8:c9:c10 -> c6:c7 -> c:c1:c2 -> c8:c9:c10 c10 :: c8:c9:c10 -> c:c1:c2 -> c8:c9:c10 c11 :: c11:c12 -> c11:c12 c12 :: c11:c12 gcd :: 0':s -> 0':s -> gcd hole_c:c1:c21_13 :: c:c1:c2 hole_0':s2_13 :: 0':s hole_c3:c4:c53_13 :: c3:c4:c5 hole_c6:c74_13 :: c6:c7 hole_c11:c125_13 :: c11:c12 hole_c8:c9:c106_13 :: c8:c9:c10 hole_gcd7_13 :: gcd gen_c:c1:c28_13 :: Nat -> c:c1:c2 gen_0':s9_13 :: Nat -> 0':s gen_c3:c4:c510_13 :: Nat -> c3:c4:c5 gen_c6:c711_13 :: Nat -> c6:c7 gen_c11:c1212_13 :: Nat -> c11:c12 gen_c8:c9:c1013_13 :: Nat -> c8:c9:c10 Generator Equations: gen_c:c1:c28_13(0) <=> c gen_c:c1:c28_13(+(x, 1)) <=> c2(gen_c:c1:c28_13(x)) gen_0':s9_13(0) <=> 0' gen_0':s9_13(+(x, 1)) <=> s(gen_0':s9_13(x)) gen_c3:c4:c510_13(0) <=> c3 gen_c3:c4:c510_13(+(x, 1)) <=> c5(gen_c3:c4:c510_13(x)) gen_c6:c711_13(0) <=> c6 gen_c6:c711_13(+(x, 1)) <=> c7(gen_c6:c711_13(x), c12) gen_c11:c1212_13(0) <=> c12 gen_c11:c1212_13(+(x, 1)) <=> c11(gen_c11:c1212_13(x)) gen_c8:c9:c1013_13(0) <=> hole_c8:c9:c106_13 gen_c8:c9:c1013_13(+(x, 1)) <=> c8(gen_c8:c9:c1013_13(x), c6, c3) The following defined symbols remain to be analysed: MIN, MAX, MINUS, any, ANY, GCD, minus, max, min, gcd They will be analysed ascendingly in the following order: MIN < GCD MAX < GCD any < MINUS ANY < MINUS MINUS < GCD any < minus minus < GCD max < GCD min < GCD minus < gcd max < gcd min < gcd ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: MIN(gen_0':s9_13(n15_13), gen_0':s9_13(n15_13)) -> gen_c:c1:c28_13(n15_13), rt in Omega(1 + n15_13) Induction Base: MIN(gen_0':s9_13(0), gen_0':s9_13(0)) ->_R^Omega(1) c Induction Step: MIN(gen_0':s9_13(+(n15_13, 1)), gen_0':s9_13(+(n15_13, 1))) ->_R^Omega(1) c2(MIN(gen_0':s9_13(n15_13), gen_0':s9_13(n15_13))) ->_IH c2(gen_c:c1:c28_13(c16_13)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (16) Complex Obligation (BEST) ---------------------------------------- (17) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: MIN(z0, 0') -> c MIN(0', z0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(z0, 0') -> c3 MAX(0', z0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(z0, 0') -> c6 MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) ANY(z0) -> c12 min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) max(z0, 0') -> z0 max(0', z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) gcd(s(z0), s(z1)) -> gcd(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))) any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 Types: MIN :: 0':s -> 0':s -> c:c1:c2 0' :: 0':s c :: c:c1:c2 c1 :: c:c1:c2 s :: 0':s -> 0':s c2 :: c:c1:c2 -> c:c1:c2 MAX :: 0':s -> 0':s -> c3:c4:c5 c3 :: c3:c4:c5 c4 :: c3:c4:c5 c5 :: c3:c4:c5 -> c3:c4:c5 MINUS :: 0':s -> 0':s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c11:c12 -> c6:c7 any :: 0':s -> 0':s ANY :: 0':s -> c11:c12 GCD :: 0':s -> 0':s -> c8:c9:c10 c8 :: c8:c9:c10 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10 minus :: 0':s -> 0':s -> 0':s max :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s c9 :: c8:c9:c10 -> c6:c7 -> c:c1:c2 -> c8:c9:c10 c10 :: c8:c9:c10 -> c:c1:c2 -> c8:c9:c10 c11 :: c11:c12 -> c11:c12 c12 :: c11:c12 gcd :: 0':s -> 0':s -> gcd hole_c:c1:c21_13 :: c:c1:c2 hole_0':s2_13 :: 0':s hole_c3:c4:c53_13 :: c3:c4:c5 hole_c6:c74_13 :: c6:c7 hole_c11:c125_13 :: c11:c12 hole_c8:c9:c106_13 :: c8:c9:c10 hole_gcd7_13 :: gcd gen_c:c1:c28_13 :: Nat -> c:c1:c2 gen_0':s9_13 :: Nat -> 0':s gen_c3:c4:c510_13 :: Nat -> c3:c4:c5 gen_c6:c711_13 :: Nat -> c6:c7 gen_c11:c1212_13 :: Nat -> c11:c12 gen_c8:c9:c1013_13 :: Nat -> c8:c9:c10 Generator Equations: gen_c:c1:c28_13(0) <=> c gen_c:c1:c28_13(+(x, 1)) <=> c2(gen_c:c1:c28_13(x)) gen_0':s9_13(0) <=> 0' gen_0':s9_13(+(x, 1)) <=> s(gen_0':s9_13(x)) gen_c3:c4:c510_13(0) <=> c3 gen_c3:c4:c510_13(+(x, 1)) <=> c5(gen_c3:c4:c510_13(x)) gen_c6:c711_13(0) <=> c6 gen_c6:c711_13(+(x, 1)) <=> c7(gen_c6:c711_13(x), c12) gen_c11:c1212_13(0) <=> c12 gen_c11:c1212_13(+(x, 1)) <=> c11(gen_c11:c1212_13(x)) gen_c8:c9:c1013_13(0) <=> hole_c8:c9:c106_13 gen_c8:c9:c1013_13(+(x, 1)) <=> c8(gen_c8:c9:c1013_13(x), c6, c3) The following defined symbols remain to be analysed: MIN, MAX, MINUS, any, ANY, GCD, minus, max, min, gcd They will be analysed ascendingly in the following order: MIN < GCD MAX < GCD any < MINUS ANY < MINUS MINUS < GCD any < minus minus < GCD max < GCD min < GCD minus < gcd max < gcd min < gcd ---------------------------------------- (18) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (19) BOUNDS(n^1, INF) ---------------------------------------- (20) Obligation: Innermost TRS: Rules: MIN(z0, 0') -> c MIN(0', z0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(z0, 0') -> c3 MAX(0', z0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(z0, 0') -> c6 MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) ANY(z0) -> c12 min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) max(z0, 0') -> z0 max(0', z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) gcd(s(z0), s(z1)) -> gcd(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))) any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 Types: MIN :: 0':s -> 0':s -> c:c1:c2 0' :: 0':s c :: c:c1:c2 c1 :: c:c1:c2 s :: 0':s -> 0':s c2 :: c:c1:c2 -> c:c1:c2 MAX :: 0':s -> 0':s -> c3:c4:c5 c3 :: c3:c4:c5 c4 :: c3:c4:c5 c5 :: c3:c4:c5 -> c3:c4:c5 MINUS :: 0':s -> 0':s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c11:c12 -> c6:c7 any :: 0':s -> 0':s ANY :: 0':s -> c11:c12 GCD :: 0':s -> 0':s -> c8:c9:c10 c8 :: c8:c9:c10 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10 minus :: 0':s -> 0':s -> 0':s max :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s c9 :: c8:c9:c10 -> c6:c7 -> c:c1:c2 -> c8:c9:c10 c10 :: c8:c9:c10 -> c:c1:c2 -> c8:c9:c10 c11 :: c11:c12 -> c11:c12 c12 :: c11:c12 gcd :: 0':s -> 0':s -> gcd hole_c:c1:c21_13 :: c:c1:c2 hole_0':s2_13 :: 0':s hole_c3:c4:c53_13 :: c3:c4:c5 hole_c6:c74_13 :: c6:c7 hole_c11:c125_13 :: c11:c12 hole_c8:c9:c106_13 :: c8:c9:c10 hole_gcd7_13 :: gcd gen_c:c1:c28_13 :: Nat -> c:c1:c2 gen_0':s9_13 :: Nat -> 0':s gen_c3:c4:c510_13 :: Nat -> c3:c4:c5 gen_c6:c711_13 :: Nat -> c6:c7 gen_c11:c1212_13 :: Nat -> c11:c12 gen_c8:c9:c1013_13 :: Nat -> c8:c9:c10 Lemmas: MIN(gen_0':s9_13(n15_13), gen_0':s9_13(n15_13)) -> gen_c:c1:c28_13(n15_13), rt in Omega(1 + n15_13) Generator Equations: gen_c:c1:c28_13(0) <=> c gen_c:c1:c28_13(+(x, 1)) <=> c2(gen_c:c1:c28_13(x)) gen_0':s9_13(0) <=> 0' gen_0':s9_13(+(x, 1)) <=> s(gen_0':s9_13(x)) gen_c3:c4:c510_13(0) <=> c3 gen_c3:c4:c510_13(+(x, 1)) <=> c5(gen_c3:c4:c510_13(x)) gen_c6:c711_13(0) <=> c6 gen_c6:c711_13(+(x, 1)) <=> c7(gen_c6:c711_13(x), c12) gen_c11:c1212_13(0) <=> c12 gen_c11:c1212_13(+(x, 1)) <=> c11(gen_c11:c1212_13(x)) gen_c8:c9:c1013_13(0) <=> hole_c8:c9:c106_13 gen_c8:c9:c1013_13(+(x, 1)) <=> c8(gen_c8:c9:c1013_13(x), c6, c3) The following defined symbols remain to be analysed: MAX, MINUS, any, ANY, GCD, minus, max, min, gcd They will be analysed ascendingly in the following order: MAX < GCD any < MINUS ANY < MINUS MINUS < GCD any < minus minus < GCD max < GCD min < GCD minus < gcd max < gcd min < gcd ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: MAX(gen_0':s9_13(n515_13), gen_0':s9_13(n515_13)) -> gen_c3:c4:c510_13(n515_13), rt in Omega(1 + n515_13) Induction Base: MAX(gen_0':s9_13(0), gen_0':s9_13(0)) ->_R^Omega(1) c3 Induction Step: MAX(gen_0':s9_13(+(n515_13, 1)), gen_0':s9_13(+(n515_13, 1))) ->_R^Omega(1) c5(MAX(gen_0':s9_13(n515_13), gen_0':s9_13(n515_13))) ->_IH c5(gen_c3:c4:c510_13(c516_13)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (22) Obligation: Innermost TRS: Rules: MIN(z0, 0') -> c MIN(0', z0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(z0, 0') -> c3 MAX(0', z0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(z0, 0') -> c6 MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) ANY(z0) -> c12 min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) max(z0, 0') -> z0 max(0', z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) gcd(s(z0), s(z1)) -> gcd(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))) any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 Types: MIN :: 0':s -> 0':s -> c:c1:c2 0' :: 0':s c :: c:c1:c2 c1 :: c:c1:c2 s :: 0':s -> 0':s c2 :: c:c1:c2 -> c:c1:c2 MAX :: 0':s -> 0':s -> c3:c4:c5 c3 :: c3:c4:c5 c4 :: c3:c4:c5 c5 :: c3:c4:c5 -> c3:c4:c5 MINUS :: 0':s -> 0':s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c11:c12 -> c6:c7 any :: 0':s -> 0':s ANY :: 0':s -> c11:c12 GCD :: 0':s -> 0':s -> c8:c9:c10 c8 :: c8:c9:c10 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10 minus :: 0':s -> 0':s -> 0':s max :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s c9 :: c8:c9:c10 -> c6:c7 -> c:c1:c2 -> c8:c9:c10 c10 :: c8:c9:c10 -> c:c1:c2 -> c8:c9:c10 c11 :: c11:c12 -> c11:c12 c12 :: c11:c12 gcd :: 0':s -> 0':s -> gcd hole_c:c1:c21_13 :: c:c1:c2 hole_0':s2_13 :: 0':s hole_c3:c4:c53_13 :: c3:c4:c5 hole_c6:c74_13 :: c6:c7 hole_c11:c125_13 :: c11:c12 hole_c8:c9:c106_13 :: c8:c9:c10 hole_gcd7_13 :: gcd gen_c:c1:c28_13 :: Nat -> c:c1:c2 gen_0':s9_13 :: Nat -> 0':s gen_c3:c4:c510_13 :: Nat -> c3:c4:c5 gen_c6:c711_13 :: Nat -> c6:c7 gen_c11:c1212_13 :: Nat -> c11:c12 gen_c8:c9:c1013_13 :: Nat -> c8:c9:c10 Lemmas: MIN(gen_0':s9_13(n15_13), gen_0':s9_13(n15_13)) -> gen_c:c1:c28_13(n15_13), rt in Omega(1 + n15_13) MAX(gen_0':s9_13(n515_13), gen_0':s9_13(n515_13)) -> gen_c3:c4:c510_13(n515_13), rt in Omega(1 + n515_13) Generator Equations: gen_c:c1:c28_13(0) <=> c gen_c:c1:c28_13(+(x, 1)) <=> c2(gen_c:c1:c28_13(x)) gen_0':s9_13(0) <=> 0' gen_0':s9_13(+(x, 1)) <=> s(gen_0':s9_13(x)) gen_c3:c4:c510_13(0) <=> c3 gen_c3:c4:c510_13(+(x, 1)) <=> c5(gen_c3:c4:c510_13(x)) gen_c6:c711_13(0) <=> c6 gen_c6:c711_13(+(x, 1)) <=> c7(gen_c6:c711_13(x), c12) gen_c11:c1212_13(0) <=> c12 gen_c11:c1212_13(+(x, 1)) <=> c11(gen_c11:c1212_13(x)) gen_c8:c9:c1013_13(0) <=> hole_c8:c9:c106_13 gen_c8:c9:c1013_13(+(x, 1)) <=> c8(gen_c8:c9:c1013_13(x), c6, c3) The following defined symbols remain to be analysed: any, MINUS, ANY, GCD, minus, max, min, gcd They will be analysed ascendingly in the following order: any < MINUS ANY < MINUS MINUS < GCD any < minus minus < GCD max < GCD min < GCD minus < gcd max < gcd min < gcd ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: any(gen_0':s9_13(+(1, n1090_13))) -> *14_13, rt in Omega(0) Induction Base: any(gen_0':s9_13(+(1, 0))) Induction Step: any(gen_0':s9_13(+(1, +(n1090_13, 1)))) ->_R^Omega(0) s(s(any(gen_0':s9_13(+(1, n1090_13))))) ->_IH s(s(*14_13)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (24) Obligation: Innermost TRS: Rules: MIN(z0, 0') -> c MIN(0', z0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(z0, 0') -> c3 MAX(0', z0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(z0, 0') -> c6 MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) ANY(z0) -> c12 min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) max(z0, 0') -> z0 max(0', z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) gcd(s(z0), s(z1)) -> gcd(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))) any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 Types: MIN :: 0':s -> 0':s -> c:c1:c2 0' :: 0':s c :: c:c1:c2 c1 :: c:c1:c2 s :: 0':s -> 0':s c2 :: c:c1:c2 -> c:c1:c2 MAX :: 0':s -> 0':s -> c3:c4:c5 c3 :: c3:c4:c5 c4 :: c3:c4:c5 c5 :: c3:c4:c5 -> c3:c4:c5 MINUS :: 0':s -> 0':s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c11:c12 -> c6:c7 any :: 0':s -> 0':s ANY :: 0':s -> c11:c12 GCD :: 0':s -> 0':s -> c8:c9:c10 c8 :: c8:c9:c10 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10 minus :: 0':s -> 0':s -> 0':s max :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s c9 :: c8:c9:c10 -> c6:c7 -> c:c1:c2 -> c8:c9:c10 c10 :: c8:c9:c10 -> c:c1:c2 -> c8:c9:c10 c11 :: c11:c12 -> c11:c12 c12 :: c11:c12 gcd :: 0':s -> 0':s -> gcd hole_c:c1:c21_13 :: c:c1:c2 hole_0':s2_13 :: 0':s hole_c3:c4:c53_13 :: c3:c4:c5 hole_c6:c74_13 :: c6:c7 hole_c11:c125_13 :: c11:c12 hole_c8:c9:c106_13 :: c8:c9:c10 hole_gcd7_13 :: gcd gen_c:c1:c28_13 :: Nat -> c:c1:c2 gen_0':s9_13 :: Nat -> 0':s gen_c3:c4:c510_13 :: Nat -> c3:c4:c5 gen_c6:c711_13 :: Nat -> c6:c7 gen_c11:c1212_13 :: Nat -> c11:c12 gen_c8:c9:c1013_13 :: Nat -> c8:c9:c10 Lemmas: MIN(gen_0':s9_13(n15_13), gen_0':s9_13(n15_13)) -> gen_c:c1:c28_13(n15_13), rt in Omega(1 + n15_13) MAX(gen_0':s9_13(n515_13), gen_0':s9_13(n515_13)) -> gen_c3:c4:c510_13(n515_13), rt in Omega(1 + n515_13) any(gen_0':s9_13(+(1, n1090_13))) -> *14_13, rt in Omega(0) Generator Equations: gen_c:c1:c28_13(0) <=> c gen_c:c1:c28_13(+(x, 1)) <=> c2(gen_c:c1:c28_13(x)) gen_0':s9_13(0) <=> 0' gen_0':s9_13(+(x, 1)) <=> s(gen_0':s9_13(x)) gen_c3:c4:c510_13(0) <=> c3 gen_c3:c4:c510_13(+(x, 1)) <=> c5(gen_c3:c4:c510_13(x)) gen_c6:c711_13(0) <=> c6 gen_c6:c711_13(+(x, 1)) <=> c7(gen_c6:c711_13(x), c12) gen_c11:c1212_13(0) <=> c12 gen_c11:c1212_13(+(x, 1)) <=> c11(gen_c11:c1212_13(x)) gen_c8:c9:c1013_13(0) <=> hole_c8:c9:c106_13 gen_c8:c9:c1013_13(+(x, 1)) <=> c8(gen_c8:c9:c1013_13(x), c6, c3) The following defined symbols remain to be analysed: ANY, MINUS, GCD, minus, max, min, gcd They will be analysed ascendingly in the following order: ANY < MINUS MINUS < GCD minus < GCD max < GCD min < GCD minus < gcd max < gcd min < gcd ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: ANY(gen_0':s9_13(+(1, n1539_13))) -> *14_13, rt in Omega(n1539_13) Induction Base: ANY(gen_0':s9_13(+(1, 0))) Induction Step: ANY(gen_0':s9_13(+(1, +(n1539_13, 1)))) ->_R^Omega(1) c11(ANY(gen_0':s9_13(+(1, n1539_13)))) ->_IH c11(*14_13) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (26) Obligation: Innermost TRS: Rules: MIN(z0, 0') -> c MIN(0', z0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(z0, 0') -> c3 MAX(0', z0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(z0, 0') -> c6 MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) ANY(z0) -> c12 min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) max(z0, 0') -> z0 max(0', z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) gcd(s(z0), s(z1)) -> gcd(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))) any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 Types: MIN :: 0':s -> 0':s -> c:c1:c2 0' :: 0':s c :: c:c1:c2 c1 :: c:c1:c2 s :: 0':s -> 0':s c2 :: c:c1:c2 -> c:c1:c2 MAX :: 0':s -> 0':s -> c3:c4:c5 c3 :: c3:c4:c5 c4 :: c3:c4:c5 c5 :: c3:c4:c5 -> c3:c4:c5 MINUS :: 0':s -> 0':s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c11:c12 -> c6:c7 any :: 0':s -> 0':s ANY :: 0':s -> c11:c12 GCD :: 0':s -> 0':s -> c8:c9:c10 c8 :: c8:c9:c10 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10 minus :: 0':s -> 0':s -> 0':s max :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s c9 :: c8:c9:c10 -> c6:c7 -> c:c1:c2 -> c8:c9:c10 c10 :: c8:c9:c10 -> c:c1:c2 -> c8:c9:c10 c11 :: c11:c12 -> c11:c12 c12 :: c11:c12 gcd :: 0':s -> 0':s -> gcd hole_c:c1:c21_13 :: c:c1:c2 hole_0':s2_13 :: 0':s hole_c3:c4:c53_13 :: c3:c4:c5 hole_c6:c74_13 :: c6:c7 hole_c11:c125_13 :: c11:c12 hole_c8:c9:c106_13 :: c8:c9:c10 hole_gcd7_13 :: gcd gen_c:c1:c28_13 :: Nat -> c:c1:c2 gen_0':s9_13 :: Nat -> 0':s gen_c3:c4:c510_13 :: Nat -> c3:c4:c5 gen_c6:c711_13 :: Nat -> c6:c7 gen_c11:c1212_13 :: Nat -> c11:c12 gen_c8:c9:c1013_13 :: Nat -> c8:c9:c10 Lemmas: MIN(gen_0':s9_13(n15_13), gen_0':s9_13(n15_13)) -> gen_c:c1:c28_13(n15_13), rt in Omega(1 + n15_13) MAX(gen_0':s9_13(n515_13), gen_0':s9_13(n515_13)) -> gen_c3:c4:c510_13(n515_13), rt in Omega(1 + n515_13) any(gen_0':s9_13(+(1, n1090_13))) -> *14_13, rt in Omega(0) ANY(gen_0':s9_13(+(1, n1539_13))) -> *14_13, rt in Omega(n1539_13) Generator Equations: gen_c:c1:c28_13(0) <=> c gen_c:c1:c28_13(+(x, 1)) <=> c2(gen_c:c1:c28_13(x)) gen_0':s9_13(0) <=> 0' gen_0':s9_13(+(x, 1)) <=> s(gen_0':s9_13(x)) gen_c3:c4:c510_13(0) <=> c3 gen_c3:c4:c510_13(+(x, 1)) <=> c5(gen_c3:c4:c510_13(x)) gen_c6:c711_13(0) <=> c6 gen_c6:c711_13(+(x, 1)) <=> c7(gen_c6:c711_13(x), c12) gen_c11:c1212_13(0) <=> c12 gen_c11:c1212_13(+(x, 1)) <=> c11(gen_c11:c1212_13(x)) gen_c8:c9:c1013_13(0) <=> hole_c8:c9:c106_13 gen_c8:c9:c1013_13(+(x, 1)) <=> c8(gen_c8:c9:c1013_13(x), c6, c3) The following defined symbols remain to be analysed: MINUS, GCD, minus, max, min, gcd They will be analysed ascendingly in the following order: MINUS < GCD minus < GCD max < GCD min < GCD minus < gcd max < gcd min < gcd ---------------------------------------- (27) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: max(gen_0':s9_13(n3375_13), gen_0':s9_13(n3375_13)) -> gen_0':s9_13(n3375_13), rt in Omega(0) Induction Base: max(gen_0':s9_13(0), gen_0':s9_13(0)) ->_R^Omega(0) gen_0':s9_13(0) Induction Step: max(gen_0':s9_13(+(n3375_13, 1)), gen_0':s9_13(+(n3375_13, 1))) ->_R^Omega(0) s(max(gen_0':s9_13(n3375_13), gen_0':s9_13(n3375_13))) ->_IH s(gen_0':s9_13(c3376_13)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (28) Obligation: Innermost TRS: Rules: MIN(z0, 0') -> c MIN(0', z0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(z0, 0') -> c3 MAX(0', z0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(z0, 0') -> c6 MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) ANY(z0) -> c12 min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) max(z0, 0') -> z0 max(0', z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) gcd(s(z0), s(z1)) -> gcd(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))) any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 Types: MIN :: 0':s -> 0':s -> c:c1:c2 0' :: 0':s c :: c:c1:c2 c1 :: c:c1:c2 s :: 0':s -> 0':s c2 :: c:c1:c2 -> c:c1:c2 MAX :: 0':s -> 0':s -> c3:c4:c5 c3 :: c3:c4:c5 c4 :: c3:c4:c5 c5 :: c3:c4:c5 -> c3:c4:c5 MINUS :: 0':s -> 0':s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c11:c12 -> c6:c7 any :: 0':s -> 0':s ANY :: 0':s -> c11:c12 GCD :: 0':s -> 0':s -> c8:c9:c10 c8 :: c8:c9:c10 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10 minus :: 0':s -> 0':s -> 0':s max :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s c9 :: c8:c9:c10 -> c6:c7 -> c:c1:c2 -> c8:c9:c10 c10 :: c8:c9:c10 -> c:c1:c2 -> c8:c9:c10 c11 :: c11:c12 -> c11:c12 c12 :: c11:c12 gcd :: 0':s -> 0':s -> gcd hole_c:c1:c21_13 :: c:c1:c2 hole_0':s2_13 :: 0':s hole_c3:c4:c53_13 :: c3:c4:c5 hole_c6:c74_13 :: c6:c7 hole_c11:c125_13 :: c11:c12 hole_c8:c9:c106_13 :: c8:c9:c10 hole_gcd7_13 :: gcd gen_c:c1:c28_13 :: Nat -> c:c1:c2 gen_0':s9_13 :: Nat -> 0':s gen_c3:c4:c510_13 :: Nat -> c3:c4:c5 gen_c6:c711_13 :: Nat -> c6:c7 gen_c11:c1212_13 :: Nat -> c11:c12 gen_c8:c9:c1013_13 :: Nat -> c8:c9:c10 Lemmas: MIN(gen_0':s9_13(n15_13), gen_0':s9_13(n15_13)) -> gen_c:c1:c28_13(n15_13), rt in Omega(1 + n15_13) MAX(gen_0':s9_13(n515_13), gen_0':s9_13(n515_13)) -> gen_c3:c4:c510_13(n515_13), rt in Omega(1 + n515_13) any(gen_0':s9_13(+(1, n1090_13))) -> *14_13, rt in Omega(0) ANY(gen_0':s9_13(+(1, n1539_13))) -> *14_13, rt in Omega(n1539_13) max(gen_0':s9_13(n3375_13), gen_0':s9_13(n3375_13)) -> gen_0':s9_13(n3375_13), rt in Omega(0) Generator Equations: gen_c:c1:c28_13(0) <=> c gen_c:c1:c28_13(+(x, 1)) <=> c2(gen_c:c1:c28_13(x)) gen_0':s9_13(0) <=> 0' gen_0':s9_13(+(x, 1)) <=> s(gen_0':s9_13(x)) gen_c3:c4:c510_13(0) <=> c3 gen_c3:c4:c510_13(+(x, 1)) <=> c5(gen_c3:c4:c510_13(x)) gen_c6:c711_13(0) <=> c6 gen_c6:c711_13(+(x, 1)) <=> c7(gen_c6:c711_13(x), c12) gen_c11:c1212_13(0) <=> c12 gen_c11:c1212_13(+(x, 1)) <=> c11(gen_c11:c1212_13(x)) gen_c8:c9:c1013_13(0) <=> hole_c8:c9:c106_13 gen_c8:c9:c1013_13(+(x, 1)) <=> c8(gen_c8:c9:c1013_13(x), c6, c3) The following defined symbols remain to be analysed: min, GCD, gcd They will be analysed ascendingly in the following order: min < GCD min < gcd ---------------------------------------- (29) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: min(gen_0':s9_13(n4421_13), gen_0':s9_13(n4421_13)) -> gen_0':s9_13(n4421_13), rt in Omega(0) Induction Base: min(gen_0':s9_13(0), gen_0':s9_13(0)) ->_R^Omega(0) 0' Induction Step: min(gen_0':s9_13(+(n4421_13, 1)), gen_0':s9_13(+(n4421_13, 1))) ->_R^Omega(0) s(min(gen_0':s9_13(n4421_13), gen_0':s9_13(n4421_13))) ->_IH s(gen_0':s9_13(c4422_13)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (30) Obligation: Innermost TRS: Rules: MIN(z0, 0') -> c MIN(0', z0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(z0, 0') -> c3 MAX(0', z0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(z0, 0') -> c6 MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) ANY(z0) -> c12 min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) max(z0, 0') -> z0 max(0', z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) gcd(s(z0), s(z1)) -> gcd(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))) any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 Types: MIN :: 0':s -> 0':s -> c:c1:c2 0' :: 0':s c :: c:c1:c2 c1 :: c:c1:c2 s :: 0':s -> 0':s c2 :: c:c1:c2 -> c:c1:c2 MAX :: 0':s -> 0':s -> c3:c4:c5 c3 :: c3:c4:c5 c4 :: c3:c4:c5 c5 :: c3:c4:c5 -> c3:c4:c5 MINUS :: 0':s -> 0':s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c11:c12 -> c6:c7 any :: 0':s -> 0':s ANY :: 0':s -> c11:c12 GCD :: 0':s -> 0':s -> c8:c9:c10 c8 :: c8:c9:c10 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10 minus :: 0':s -> 0':s -> 0':s max :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s c9 :: c8:c9:c10 -> c6:c7 -> c:c1:c2 -> c8:c9:c10 c10 :: c8:c9:c10 -> c:c1:c2 -> c8:c9:c10 c11 :: c11:c12 -> c11:c12 c12 :: c11:c12 gcd :: 0':s -> 0':s -> gcd hole_c:c1:c21_13 :: c:c1:c2 hole_0':s2_13 :: 0':s hole_c3:c4:c53_13 :: c3:c4:c5 hole_c6:c74_13 :: c6:c7 hole_c11:c125_13 :: c11:c12 hole_c8:c9:c106_13 :: c8:c9:c10 hole_gcd7_13 :: gcd gen_c:c1:c28_13 :: Nat -> c:c1:c2 gen_0':s9_13 :: Nat -> 0':s gen_c3:c4:c510_13 :: Nat -> c3:c4:c5 gen_c6:c711_13 :: Nat -> c6:c7 gen_c11:c1212_13 :: Nat -> c11:c12 gen_c8:c9:c1013_13 :: Nat -> c8:c9:c10 Lemmas: MIN(gen_0':s9_13(n15_13), gen_0':s9_13(n15_13)) -> gen_c:c1:c28_13(n15_13), rt in Omega(1 + n15_13) MAX(gen_0':s9_13(n515_13), gen_0':s9_13(n515_13)) -> gen_c3:c4:c510_13(n515_13), rt in Omega(1 + n515_13) any(gen_0':s9_13(+(1, n1090_13))) -> *14_13, rt in Omega(0) ANY(gen_0':s9_13(+(1, n1539_13))) -> *14_13, rt in Omega(n1539_13) max(gen_0':s9_13(n3375_13), gen_0':s9_13(n3375_13)) -> gen_0':s9_13(n3375_13), rt in Omega(0) min(gen_0':s9_13(n4421_13), gen_0':s9_13(n4421_13)) -> gen_0':s9_13(n4421_13), rt in Omega(0) Generator Equations: gen_c:c1:c28_13(0) <=> c gen_c:c1:c28_13(+(x, 1)) <=> c2(gen_c:c1:c28_13(x)) gen_0':s9_13(0) <=> 0' gen_0':s9_13(+(x, 1)) <=> s(gen_0':s9_13(x)) gen_c3:c4:c510_13(0) <=> c3 gen_c3:c4:c510_13(+(x, 1)) <=> c5(gen_c3:c4:c510_13(x)) gen_c6:c711_13(0) <=> c6 gen_c6:c711_13(+(x, 1)) <=> c7(gen_c6:c711_13(x), c12) gen_c11:c1212_13(0) <=> c12 gen_c11:c1212_13(+(x, 1)) <=> c11(gen_c11:c1212_13(x)) gen_c8:c9:c1013_13(0) <=> hole_c8:c9:c106_13 gen_c8:c9:c1013_13(+(x, 1)) <=> c8(gen_c8:c9:c1013_13(x), c6, c3) The following defined symbols remain to be analysed: GCD, gcd ---------------------------------------- (31) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) gcd(s(z0), s(z1)) -> gcd(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))) any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 Tuples: MIN(z0, 0) -> c MIN(0, z0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(z0, 0) -> c3 MAX(0, z0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(z0, 0) -> c6 MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) ANY(z0) -> c12 S tuples: MIN(z0, 0) -> c MIN(0, z0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(z0, 0) -> c3 MAX(0, z0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(z0, 0) -> c6 MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) ANY(z0) -> c12 K tuples:none Defined Rule Symbols: min_2, max_2, minus_2, gcd_2, any_1 Defined Pair Symbols: MIN_2, MAX_2, MINUS_2, GCD_2, ANY_1 Compound Symbols: c, c1, c2_1, c3, c4, c5_1, c6, c7_2, c8_3, c9_3, c10_2, c11_1, c12 ---------------------------------------- (33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing nodes: MINUS(z0, 0) -> c6 MIN(z0, 0) -> c MIN(0, z0) -> c1 MAX(0, z0) -> c4 ANY(z0) -> c12 MAX(z0, 0) -> c3 ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) gcd(s(z0), s(z1)) -> gcd(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))) any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) K tuples:none Defined Rule Symbols: min_2, max_2, minus_2, gcd_2, any_1 Defined Pair Symbols: MIN_2, MAX_2, MINUS_2, GCD_2, ANY_1 Compound Symbols: c2_1, c5_1, c7_2, c8_3, c9_3, c10_2, c11_1 ---------------------------------------- (35) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: gcd(s(z0), s(z1)) -> gcd(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, MINUS_2, GCD_2, ANY_1 Compound Symbols: c2_1, c5_1, c7_2, c8_3, c9_3, c10_2, c11_1 ---------------------------------------- (37) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (38) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) The (relative) TRS S consists of the following rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Rewrite Strategy: INNERMOST ---------------------------------------- (39) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (40) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (41) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (42) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) [1] MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) [1] MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) [1] GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) [1] GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) [1] GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) [1] ANY(s(z0)) -> c11(ANY(z0)) [1] any(s(z0)) -> s(s(any(z0))) [0] any(z0) -> z0 [0] minus(z0, 0) -> z0 [0] minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) [0] max(z0, 0) -> z0 [0] max(0, z0) -> z0 [0] max(s(z0), s(z1)) -> s(max(z0, z1)) [0] min(z0, 0) -> 0 [0] min(0, z0) -> 0 [0] min(s(z0), s(z1)) -> s(min(z0, z1)) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (43) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (44) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) [1] MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) [1] MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) [1] GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) [1] GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) [1] GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) [1] ANY(s(z0)) -> c11(ANY(z0)) [1] any(s(z0)) -> s(s(any(z0))) [0] any(z0) -> z0 [0] minus(z0, 0) -> z0 [0] minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) [0] max(z0, 0) -> z0 [0] max(0, z0) -> z0 [0] max(s(z0), s(z1)) -> s(max(z0, z1)) [0] min(z0, 0) -> 0 [0] min(0, z0) -> 0 [0] min(s(z0), s(z1)) -> s(min(z0, z1)) [0] The TRS has the following type information: MIN :: s:0 -> s:0 -> c2 s :: s:0 -> s:0 c2 :: c2 -> c2 MAX :: s:0 -> s:0 -> c5 c5 :: c5 -> c5 MINUS :: s:0 -> s:0 -> c7 c7 :: c7 -> c11 -> c7 any :: s:0 -> s:0 ANY :: s:0 -> c11 GCD :: s:0 -> s:0 -> c8:c9:c10 c8 :: c8:c9:c10 -> c7 -> c5 -> c8:c9:c10 minus :: s:0 -> s:0 -> s:0 max :: s:0 -> s:0 -> s:0 min :: s:0 -> s:0 -> s:0 c9 :: c8:c9:c10 -> c7 -> c2 -> c8:c9:c10 c10 :: c8:c9:c10 -> c2 -> c8:c9:c10 c11 :: c11 -> c11 0 :: s:0 Rewrite Strategy: INNERMOST ---------------------------------------- (45) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: MIN_2 MAX_2 MINUS_2 GCD_2 ANY_1 (c) The following functions are completely defined: any_1 minus_2 max_2 min_2 Due to the following rules being added: any(v0) -> 0 [0] minus(v0, v1) -> 0 [0] max(v0, v1) -> 0 [0] min(v0, v1) -> 0 [0] And the following fresh constants: const, const1, const2, const3, const4 ---------------------------------------- (46) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) [1] MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) [1] MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) [1] GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) [1] GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) [1] GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) [1] ANY(s(z0)) -> c11(ANY(z0)) [1] any(s(z0)) -> s(s(any(z0))) [0] any(z0) -> z0 [0] minus(z0, 0) -> z0 [0] minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) [0] max(z0, 0) -> z0 [0] max(0, z0) -> z0 [0] max(s(z0), s(z1)) -> s(max(z0, z1)) [0] min(z0, 0) -> 0 [0] min(0, z0) -> 0 [0] min(s(z0), s(z1)) -> s(min(z0, z1)) [0] any(v0) -> 0 [0] minus(v0, v1) -> 0 [0] max(v0, v1) -> 0 [0] min(v0, v1) -> 0 [0] The TRS has the following type information: MIN :: s:0 -> s:0 -> c2 s :: s:0 -> s:0 c2 :: c2 -> c2 MAX :: s:0 -> s:0 -> c5 c5 :: c5 -> c5 MINUS :: s:0 -> s:0 -> c7 c7 :: c7 -> c11 -> c7 any :: s:0 -> s:0 ANY :: s:0 -> c11 GCD :: s:0 -> s:0 -> c8:c9:c10 c8 :: c8:c9:c10 -> c7 -> c5 -> c8:c9:c10 minus :: s:0 -> s:0 -> s:0 max :: s:0 -> s:0 -> s:0 min :: s:0 -> s:0 -> s:0 c9 :: c8:c9:c10 -> c7 -> c2 -> c8:c9:c10 c10 :: c8:c9:c10 -> c2 -> c8:c9:c10 c11 :: c11 -> c11 0 :: s:0 const :: c2 const1 :: c5 const2 :: c7 const3 :: c11 const4 :: c8:c9:c10 Rewrite Strategy: INNERMOST ---------------------------------------- (47) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (48) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) [1] MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) [1] MINUS(s(z0), s(s(z0'))) -> c7(MINUS(z0, s(s(any(z0')))), ANY(s(z0'))) [1] MINUS(s(z0), s(z1)) -> c7(MINUS(z0, z1), ANY(z1)) [1] MINUS(s(z0), s(z1)) -> c7(MINUS(z0, 0), ANY(z1)) [1] GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) [1] GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) [1] GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) [1] ANY(s(z0)) -> c11(ANY(z0)) [1] any(s(z0)) -> s(s(any(z0))) [0] any(z0) -> z0 [0] minus(z0, 0) -> z0 [0] minus(s(z0), s(s(z013))) -> s(minus(z0, s(s(any(z013))))) [0] minus(s(z0), s(z1)) -> s(minus(z0, z1)) [0] minus(s(z0), s(z1)) -> s(minus(z0, 0)) [0] max(z0, 0) -> z0 [0] max(0, z0) -> z0 [0] max(s(z0), s(z1)) -> s(max(z0, z1)) [0] min(z0, 0) -> 0 [0] min(0, z0) -> 0 [0] min(s(z0), s(z1)) -> s(min(z0, z1)) [0] any(v0) -> 0 [0] minus(v0, v1) -> 0 [0] max(v0, v1) -> 0 [0] min(v0, v1) -> 0 [0] The TRS has the following type information: MIN :: s:0 -> s:0 -> c2 s :: s:0 -> s:0 c2 :: c2 -> c2 MAX :: s:0 -> s:0 -> c5 c5 :: c5 -> c5 MINUS :: s:0 -> s:0 -> c7 c7 :: c7 -> c11 -> c7 any :: s:0 -> s:0 ANY :: s:0 -> c11 GCD :: s:0 -> s:0 -> c8:c9:c10 c8 :: c8:c9:c10 -> c7 -> c5 -> c8:c9:c10 minus :: s:0 -> s:0 -> s:0 max :: s:0 -> s:0 -> s:0 min :: s:0 -> s:0 -> s:0 c9 :: c8:c9:c10 -> c7 -> c2 -> c8:c9:c10 c10 :: c8:c9:c10 -> c2 -> c8:c9:c10 c11 :: c11 -> c11 0 :: s:0 const :: c2 const1 :: c5 const2 :: c7 const3 :: c11 const4 :: c8:c9:c10 Rewrite Strategy: INNERMOST ---------------------------------------- (49) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 const => 0 const1 => 0 const2 => 0 const3 => 0 const4 => 0 ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ 1 }-> 1 + ANY(z0) :|: z = 1 + z0, z0 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z0, z1), min(z0, z1)), 1 + min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z0, z1), min(z0, z1)), 1 + min(z0, z1)) + MINUS(max(z0, z1), min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z0, z1), min(z0, z1)), 1 + min(z0, z1)) + MINUS(max(z0, z1), min(z0, z1)) + MAX(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 MAX(z, z') -{ 1 }-> 1 + MAX(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 MIN(z, z') -{ 1 }-> 1 + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 MINUS(z, z') -{ 1 }-> 1 + MINUS(z0, z1) + ANY(z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 MINUS(z, z') -{ 1 }-> 1 + MINUS(z0, 0) + ANY(z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 MINUS(z, z') -{ 1 }-> 1 + MINUS(z0, 1 + (1 + any(z0'))) + ANY(1 + z0') :|: z' = 1 + (1 + z0'), z0' >= 0, z = 1 + z0, z0 >= 0 any(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0 any(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 any(z) -{ 0 }-> 1 + (1 + any(z0)) :|: z = 1 + z0, z0 >= 0 max(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0 max(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0 max(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 max(z, z') -{ 0 }-> 1 + max(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 min(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0 min(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 min(z, z') -{ 0 }-> 1 + min(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 minus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 minus(z, z') -{ 0 }-> 1 + minus(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 minus(z, z') -{ 0 }-> 1 + minus(z0, 0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 minus(z, z') -{ 0 }-> 1 + minus(z0, 1 + (1 + any(z013))) :|: z' = 1 + (1 + z013), z013 >= 0, z = 1 + z0, z0 >= 0 ---------------------------------------- (51) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ 1 }-> 1 + ANY(z - 1) :|: z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, 0) + ANY(z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) + ANY(z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, 1 + (1 + any(z' - 2))) + ANY(1 + (z' - 2)) :|: z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0 ---------------------------------------- (53) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { ANY } { MIN } { max } { any } { min } { MAX } { MINUS } { minus } { GCD } ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ 1 }-> 1 + ANY(z - 1) :|: z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, 0) + ANY(z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) + ANY(z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, 1 + (1 + any(z' - 2))) + ANY(1 + (z' - 2)) :|: z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {ANY}, {MIN}, {max}, {any}, {min}, {MAX}, {MINUS}, {minus}, {GCD} ---------------------------------------- (55) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ 1 }-> 1 + ANY(z - 1) :|: z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, 0) + ANY(z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) + ANY(z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, 1 + (1 + any(z' - 2))) + ANY(1 + (z' - 2)) :|: z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {ANY}, {MIN}, {max}, {any}, {min}, {MAX}, {MINUS}, {minus}, {GCD} ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: ANY after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ 1 }-> 1 + ANY(z - 1) :|: z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, 0) + ANY(z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) + ANY(z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, 1 + (1 + any(z' - 2))) + ANY(1 + (z' - 2)) :|: z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {ANY}, {MIN}, {max}, {any}, {min}, {MAX}, {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: ?, size: O(1) [0] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: ANY after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ 1 }-> 1 + ANY(z - 1) :|: z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, 0) + ANY(z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) + ANY(z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, 1 + (1 + any(z' - 2))) + ANY(1 + (z' - 2)) :|: z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {MIN}, {max}, {any}, {min}, {MAX}, {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] ---------------------------------------- (61) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 0) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, z' - 1) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 1 + (1 + any(z' - 2))) + s :|: s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {MIN}, {max}, {any}, {min}, {MAX}, {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: MIN after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 0) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, z' - 1) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 1 + (1 + any(z' - 2))) + s :|: s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {MIN}, {max}, {any}, {min}, {MAX}, {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: ?, size: O(1) [0] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: MIN after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 0) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, z' - 1) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 1 + (1 + any(z' - 2))) + s :|: s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {max}, {any}, {min}, {MAX}, {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] ---------------------------------------- (67) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 0) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, z' - 1) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 1 + (1 + any(z' - 2))) + s :|: s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {max}, {any}, {min}, {MAX}, {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: max after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 0) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, z' - 1) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 1 + (1 + any(z' - 2))) + s :|: s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {max}, {any}, {min}, {MAX}, {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (71) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: max after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 0) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, z' - 1) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 1 + (1 + any(z' - 2))) + s :|: s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {any}, {min}, {MAX}, {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] ---------------------------------------- (73) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s9, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + s4 :|: s9 >= 0, s9 <= z' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(s5, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(s6, min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s7, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(s8, min(z - 1, z' - 1)) + s3 :|: s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 0) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, z' - 1) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 1 + (1 + any(z' - 2))) + s :|: s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + s10 :|: s10 >= 0, s10 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {any}, {min}, {MAX}, {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] ---------------------------------------- (75) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: any after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s9, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + s4 :|: s9 >= 0, s9 <= z' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(s5, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(s6, min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s7, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(s8, min(z - 1, z' - 1)) + s3 :|: s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 0) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, z' - 1) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 1 + (1 + any(z' - 2))) + s :|: s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + s10 :|: s10 >= 0, s10 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {any}, {min}, {MAX}, {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] any: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (77) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: any after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s9, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + s4 :|: s9 >= 0, s9 <= z' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(s5, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(s6, min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s7, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(s8, min(z - 1, z' - 1)) + s3 :|: s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 0) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, z' - 1) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 1 + (1 + any(z' - 2))) + s :|: s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + s10 :|: s10 >= 0, s10 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {min}, {MAX}, {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] any: runtime: O(1) [0], size: O(n^1) [2*z] ---------------------------------------- (79) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (80) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s9, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + s4 :|: s9 >= 0, s9 <= z' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(s5, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(s6, min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s7, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(s8, min(z - 1, z' - 1)) + s3 :|: s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 0) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, z' - 1) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 1 + (1 + s11)) + s :|: s11 >= 0, s11 <= 2 * (z' - 2), s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + s12) :|: s12 >= 0, s12 <= 2 * (z - 1), z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + s10 :|: s10 >= 0, s10 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + s13)) :|: s13 >= 0, s13 <= 2 * (z' - 2), z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {min}, {MAX}, {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] any: runtime: O(1) [0], size: O(n^1) [2*z] ---------------------------------------- (81) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: min after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (82) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s9, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + s4 :|: s9 >= 0, s9 <= z' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(s5, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(s6, min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s7, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(s8, min(z - 1, z' - 1)) + s3 :|: s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 0) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, z' - 1) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 1 + (1 + s11)) + s :|: s11 >= 0, s11 <= 2 * (z' - 2), s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + s12) :|: s12 >= 0, s12 <= 2 * (z - 1), z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + s10 :|: s10 >= 0, s10 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + s13)) :|: s13 >= 0, s13 <= 2 * (z' - 2), z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {min}, {MAX}, {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] any: runtime: O(1) [0], size: O(n^1) [2*z] min: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (83) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: min after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (84) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s9, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + s4 :|: s9 >= 0, s9 <= z' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(s5, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(s6, min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s7, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1)) + MINUS(s8, min(z - 1, z' - 1)) + s3 :|: s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 0) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, z' - 1) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 1 + (1 + s11)) + s :|: s11 >= 0, s11 <= 2 * (z' - 2), s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + s12) :|: s12 >= 0, s12 <= 2 * (z - 1), z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + s10 :|: s10 >= 0, s10 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + s13)) :|: s13 >= 0, s13 <= 2 * (z' - 2), z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {MAX}, {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] any: runtime: O(1) [0], size: O(n^1) [2*z] min: runtime: O(1) [0], size: O(n^1) [z'] ---------------------------------------- (85) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (86) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s9, s20), 1 + s21) + s4 :|: s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= z' - 1, s9 >= 0, s9 <= z' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(s5, s14), 1 + s15) + MINUS(s6, s16) + MAX(z - 1, z' - 1) :|: s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= z' - 1, s16 >= 0, s16 <= z' - 1, s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s7, s17), 1 + s18) + MINUS(s8, s19) + s3 :|: s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z' - 1, s19 >= 0, s19 <= z' - 1, s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 0) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, z' - 1) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 1 + (1 + s11)) + s :|: s11 >= 0, s11 <= 2 * (z' - 2), s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + s12) :|: s12 >= 0, s12 <= 2 * (z - 1), z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + s10 :|: s10 >= 0, s10 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + s22 :|: s22 >= 0, s22 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + s13)) :|: s13 >= 0, s13 <= 2 * (z' - 2), z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {MAX}, {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] any: runtime: O(1) [0], size: O(n^1) [2*z] min: runtime: O(1) [0], size: O(n^1) [z'] ---------------------------------------- (87) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: MAX after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (88) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s9, s20), 1 + s21) + s4 :|: s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= z' - 1, s9 >= 0, s9 <= z' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(s5, s14), 1 + s15) + MINUS(s6, s16) + MAX(z - 1, z' - 1) :|: s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= z' - 1, s16 >= 0, s16 <= z' - 1, s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s7, s17), 1 + s18) + MINUS(s8, s19) + s3 :|: s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z' - 1, s19 >= 0, s19 <= z' - 1, s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 0) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, z' - 1) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 1 + (1 + s11)) + s :|: s11 >= 0, s11 <= 2 * (z' - 2), s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + s12) :|: s12 >= 0, s12 <= 2 * (z - 1), z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + s10 :|: s10 >= 0, s10 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + s22 :|: s22 >= 0, s22 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + s13)) :|: s13 >= 0, s13 <= 2 * (z' - 2), z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {MAX}, {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] any: runtime: O(1) [0], size: O(n^1) [2*z] min: runtime: O(1) [0], size: O(n^1) [z'] MAX: runtime: ?, size: O(1) [0] ---------------------------------------- (89) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: MAX after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (90) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s9, s20), 1 + s21) + s4 :|: s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= z' - 1, s9 >= 0, s9 <= z' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ 1 }-> 1 + GCD(minus(s5, s14), 1 + s15) + MINUS(s6, s16) + MAX(z - 1, z' - 1) :|: s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= z' - 1, s16 >= 0, s16 <= z' - 1, s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s7, s17), 1 + s18) + MINUS(s8, s19) + s3 :|: s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z' - 1, s19 >= 0, s19 <= z' - 1, s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 0) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, z' - 1) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 1 + (1 + s11)) + s :|: s11 >= 0, s11 <= 2 * (z' - 2), s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + s12) :|: s12 >= 0, s12 <= 2 * (z - 1), z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + s10 :|: s10 >= 0, s10 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + s22 :|: s22 >= 0, s22 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + s13)) :|: s13 >= 0, s13 <= 2 * (z' - 2), z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] any: runtime: O(1) [0], size: O(n^1) [2*z] min: runtime: O(1) [0], size: O(n^1) [z'] MAX: runtime: O(n^1) [z'], size: O(1) [0] ---------------------------------------- (91) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (92) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s9, s20), 1 + s21) + s4 :|: s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= z' - 1, s9 >= 0, s9 <= z' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s5, s14), 1 + s15) + MINUS(s6, s16) + s24 :|: s24 >= 0, s24 <= 0, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= z' - 1, s16 >= 0, s16 <= z' - 1, s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s7, s17), 1 + s18) + MINUS(s8, s19) + s3 :|: s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z' - 1, s19 >= 0, s19 <= z' - 1, s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ z' }-> 1 + s23 :|: s23 >= 0, s23 <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 0) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, z' - 1) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 1 + (1 + s11)) + s :|: s11 >= 0, s11 <= 2 * (z' - 2), s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + s12) :|: s12 >= 0, s12 <= 2 * (z - 1), z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + s10 :|: s10 >= 0, s10 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + s22 :|: s22 >= 0, s22 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + s13)) :|: s13 >= 0, s13 <= 2 * (z' - 2), z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] any: runtime: O(1) [0], size: O(n^1) [2*z] min: runtime: O(1) [0], size: O(n^1) [z'] MAX: runtime: O(n^1) [z'], size: O(1) [0] ---------------------------------------- (93) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: MINUS after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (94) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s9, s20), 1 + s21) + s4 :|: s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= z' - 1, s9 >= 0, s9 <= z' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s5, s14), 1 + s15) + MINUS(s6, s16) + s24 :|: s24 >= 0, s24 <= 0, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= z' - 1, s16 >= 0, s16 <= z' - 1, s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s7, s17), 1 + s18) + MINUS(s8, s19) + s3 :|: s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z' - 1, s19 >= 0, s19 <= z' - 1, s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ z' }-> 1 + s23 :|: s23 >= 0, s23 <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 0) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, z' - 1) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 1 + (1 + s11)) + s :|: s11 >= 0, s11 <= 2 * (z' - 2), s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + s12) :|: s12 >= 0, s12 <= 2 * (z - 1), z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + s10 :|: s10 >= 0, s10 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + s22 :|: s22 >= 0, s22 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + s13)) :|: s13 >= 0, s13 <= 2 * (z' - 2), z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {MINUS}, {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] any: runtime: O(1) [0], size: O(n^1) [2*z] min: runtime: O(1) [0], size: O(n^1) [z'] MAX: runtime: O(n^1) [z'], size: O(1) [0] MINUS: runtime: ?, size: O(1) [0] ---------------------------------------- (95) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: MINUS after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (96) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s9, s20), 1 + s21) + s4 :|: s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= z' - 1, s9 >= 0, s9 <= z' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s5, s14), 1 + s15) + MINUS(s6, s16) + s24 :|: s24 >= 0, s24 <= 0, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= z' - 1, s16 >= 0, s16 <= z' - 1, s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s7, s17), 1 + s18) + MINUS(s8, s19) + s3 :|: s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z' - 1, s19 >= 0, s19 <= z' - 1, s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ z' }-> 1 + s23 :|: s23 >= 0, s23 <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 0) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, z' - 1) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ z' }-> 1 + MINUS(z - 1, 1 + (1 + s11)) + s :|: s11 >= 0, s11 <= 2 * (z' - 2), s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + s12) :|: s12 >= 0, s12 <= 2 * (z - 1), z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + s10 :|: s10 >= 0, s10 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + s22 :|: s22 >= 0, s22 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + s13)) :|: s13 >= 0, s13 <= 2 * (z' - 2), z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] any: runtime: O(1) [0], size: O(n^1) [2*z] min: runtime: O(1) [0], size: O(n^1) [z'] MAX: runtime: O(n^1) [z'], size: O(1) [0] MINUS: runtime: EXP, size: O(1) [0] ---------------------------------------- (97) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (98) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s9, s20), 1 + s21) + s4 :|: s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= z' - 1, s9 >= 0, s9 <= z' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ inf1 + z' }-> 1 + GCD(minus(s5, s14), 1 + s15) + s28 + s24 :|: s28 >= 0, s28 <= 0, s24 >= 0, s24 <= 0, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= z' - 1, s16 >= 0, s16 <= z' - 1, s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ inf2 + z' }-> 1 + GCD(minus(s7, s17), 1 + s18) + s29 + s3 :|: s29 >= 0, s29 <= 0, s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z' - 1, s19 >= 0, s19 <= z' - 1, s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ z' }-> 1 + s23 :|: s23 >= 0, s23 <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ inf + z' }-> 1 + s25 + s :|: s25 >= 0, s25 <= 0, s11 >= 0, s11 <= 2 * (z' - 2), s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 MINUS(z, z') -{ inf' + z' }-> 1 + s26 + s' :|: s26 >= 0, s26 <= 0, s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ inf'' + z' }-> 1 + s27 + s'' :|: s27 >= 0, s27 <= 0, s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + s12) :|: s12 >= 0, s12 <= 2 * (z - 1), z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + s10 :|: s10 >= 0, s10 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + s22 :|: s22 >= 0, s22 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + s13)) :|: s13 >= 0, s13 <= 2 * (z' - 2), z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] any: runtime: O(1) [0], size: O(n^1) [2*z] min: runtime: O(1) [0], size: O(n^1) [z'] MAX: runtime: O(n^1) [z'], size: O(1) [0] MINUS: runtime: EXP, size: O(1) [0] ---------------------------------------- (99) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (100) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s9, s20), 1 + s21) + s4 :|: s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= z' - 1, s9 >= 0, s9 <= z' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ inf1 + z' }-> 1 + GCD(minus(s5, s14), 1 + s15) + s28 + s24 :|: s28 >= 0, s28 <= 0, s24 >= 0, s24 <= 0, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= z' - 1, s16 >= 0, s16 <= z' - 1, s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ inf2 + z' }-> 1 + GCD(minus(s7, s17), 1 + s18) + s29 + s3 :|: s29 >= 0, s29 <= 0, s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z' - 1, s19 >= 0, s19 <= z' - 1, s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ z' }-> 1 + s23 :|: s23 >= 0, s23 <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ inf + z' }-> 1 + s25 + s :|: s25 >= 0, s25 <= 0, s11 >= 0, s11 <= 2 * (z' - 2), s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 MINUS(z, z') -{ inf' + z' }-> 1 + s26 + s' :|: s26 >= 0, s26 <= 0, s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ inf'' + z' }-> 1 + s27 + s'' :|: s27 >= 0, s27 <= 0, s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + s12) :|: s12 >= 0, s12 <= 2 * (z - 1), z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + s10 :|: s10 >= 0, s10 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + s22 :|: s22 >= 0, s22 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + s13)) :|: s13 >= 0, s13 <= 2 * (z' - 2), z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {minus}, {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] any: runtime: O(1) [0], size: O(n^1) [2*z] min: runtime: O(1) [0], size: O(n^1) [z'] MAX: runtime: O(n^1) [z'], size: O(1) [0] MINUS: runtime: EXP, size: O(1) [0] minus: runtime: ?, size: O(n^1) [z] ---------------------------------------- (101) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (102) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(minus(s9, s20), 1 + s21) + s4 :|: s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= z' - 1, s9 >= 0, s9 <= z' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ inf1 + z' }-> 1 + GCD(minus(s5, s14), 1 + s15) + s28 + s24 :|: s28 >= 0, s28 <= 0, s24 >= 0, s24 <= 0, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= z' - 1, s16 >= 0, s16 <= z' - 1, s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ inf2 + z' }-> 1 + GCD(minus(s7, s17), 1 + s18) + s29 + s3 :|: s29 >= 0, s29 <= 0, s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z' - 1, s19 >= 0, s19 <= z' - 1, s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ z' }-> 1 + s23 :|: s23 >= 0, s23 <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ inf + z' }-> 1 + s25 + s :|: s25 >= 0, s25 <= 0, s11 >= 0, s11 <= 2 * (z' - 2), s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 MINUS(z, z') -{ inf' + z' }-> 1 + s26 + s' :|: s26 >= 0, s26 <= 0, s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ inf'' + z' }-> 1 + s27 + s'' :|: s27 >= 0, s27 <= 0, s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + s12) :|: s12 >= 0, s12 <= 2 * (z - 1), z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + s10 :|: s10 >= 0, s10 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + s22 :|: s22 >= 0, s22 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + minus(z - 1, 1 + (1 + s13)) :|: s13 >= 0, s13 <= 2 * (z' - 2), z' - 2 >= 0, z - 1 >= 0 Function symbols to be analyzed: {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] any: runtime: O(1) [0], size: O(n^1) [2*z] min: runtime: O(1) [0], size: O(n^1) [z'] MAX: runtime: O(n^1) [z'], size: O(1) [0] MINUS: runtime: EXP, size: O(1) [0] minus: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (103) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (104) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(s32, 1 + s21) + s4 :|: s32 >= 0, s32 <= s9, s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= z' - 1, s9 >= 0, s9 <= z' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ inf1 + z' }-> 1 + GCD(s30, 1 + s15) + s28 + s24 :|: s30 >= 0, s30 <= s5, s28 >= 0, s28 <= 0, s24 >= 0, s24 <= 0, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= z' - 1, s16 >= 0, s16 <= z' - 1, s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ inf2 + z' }-> 1 + GCD(s31, 1 + s18) + s29 + s3 :|: s31 >= 0, s31 <= s7, s29 >= 0, s29 <= 0, s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z' - 1, s19 >= 0, s19 <= z' - 1, s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ z' }-> 1 + s23 :|: s23 >= 0, s23 <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ inf + z' }-> 1 + s25 + s :|: s25 >= 0, s25 <= 0, s11 >= 0, s11 <= 2 * (z' - 2), s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 MINUS(z, z') -{ inf' + z' }-> 1 + s26 + s' :|: s26 >= 0, s26 <= 0, s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ inf'' + z' }-> 1 + s27 + s'' :|: s27 >= 0, s27 <= 0, s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + s12) :|: s12 >= 0, s12 <= 2 * (z - 1), z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + s10 :|: s10 >= 0, s10 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + s22 :|: s22 >= 0, s22 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + s33 :|: s33 >= 0, s33 <= z - 1, s13 >= 0, s13 <= 2 * (z' - 2), z' - 2 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + s34 :|: s34 >= 0, s34 <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + s35 :|: s35 >= 0, s35 <= z - 1, z' - 1 >= 0, z - 1 >= 0 Function symbols to be analyzed: {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] any: runtime: O(1) [0], size: O(n^1) [2*z] min: runtime: O(1) [0], size: O(n^1) [z'] MAX: runtime: O(n^1) [z'], size: O(1) [0] MINUS: runtime: EXP, size: O(1) [0] minus: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (105) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: GCD after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (106) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(s32, 1 + s21) + s4 :|: s32 >= 0, s32 <= s9, s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= z' - 1, s9 >= 0, s9 <= z' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ inf1 + z' }-> 1 + GCD(s30, 1 + s15) + s28 + s24 :|: s30 >= 0, s30 <= s5, s28 >= 0, s28 <= 0, s24 >= 0, s24 <= 0, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= z' - 1, s16 >= 0, s16 <= z' - 1, s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ inf2 + z' }-> 1 + GCD(s31, 1 + s18) + s29 + s3 :|: s31 >= 0, s31 <= s7, s29 >= 0, s29 <= 0, s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z' - 1, s19 >= 0, s19 <= z' - 1, s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ z' }-> 1 + s23 :|: s23 >= 0, s23 <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ inf + z' }-> 1 + s25 + s :|: s25 >= 0, s25 <= 0, s11 >= 0, s11 <= 2 * (z' - 2), s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 MINUS(z, z') -{ inf' + z' }-> 1 + s26 + s' :|: s26 >= 0, s26 <= 0, s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ inf'' + z' }-> 1 + s27 + s'' :|: s27 >= 0, s27 <= 0, s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + s12) :|: s12 >= 0, s12 <= 2 * (z - 1), z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + s10 :|: s10 >= 0, s10 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + s22 :|: s22 >= 0, s22 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + s33 :|: s33 >= 0, s33 <= z - 1, s13 >= 0, s13 <= 2 * (z' - 2), z' - 2 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + s34 :|: s34 >= 0, s34 <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + s35 :|: s35 >= 0, s35 <= z - 1, z' - 1 >= 0, z - 1 >= 0 Function symbols to be analyzed: {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] any: runtime: O(1) [0], size: O(n^1) [2*z] min: runtime: O(1) [0], size: O(n^1) [z'] MAX: runtime: O(n^1) [z'], size: O(1) [0] MINUS: runtime: EXP, size: O(1) [0] minus: runtime: O(1) [0], size: O(n^1) [z] GCD: runtime: ?, size: O(1) [0] ---------------------------------------- (107) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: GCD after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (108) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 GCD(z, z') -{ z' }-> 1 + GCD(s32, 1 + s21) + s4 :|: s32 >= 0, s32 <= s9, s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= z' - 1, s9 >= 0, s9 <= z' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ inf1 + z' }-> 1 + GCD(s30, 1 + s15) + s28 + s24 :|: s30 >= 0, s30 <= s5, s28 >= 0, s28 <= 0, s24 >= 0, s24 <= 0, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= z' - 1, s16 >= 0, s16 <= z' - 1, s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 GCD(z, z') -{ inf2 + z' }-> 1 + GCD(s31, 1 + s18) + s29 + s3 :|: s31 >= 0, s31 <= s7, s29 >= 0, s29 <= 0, s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z' - 1, s19 >= 0, s19 <= z' - 1, s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0 MAX(z, z') -{ z' }-> 1 + s23 :|: s23 >= 0, s23 <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ inf + z' }-> 1 + s25 + s :|: s25 >= 0, s25 <= 0, s11 >= 0, s11 <= 2 * (z' - 2), s >= 0, s <= 0, z' - 2 >= 0, z - 1 >= 0 MINUS(z, z') -{ inf' + z' }-> 1 + s26 + s' :|: s26 >= 0, s26 <= 0, s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ inf'' + z' }-> 1 + s27 + s'' :|: s27 >= 0, s27 <= 0, s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 any(z) -{ 0 }-> z :|: z >= 0 any(z) -{ 0 }-> 0 :|: z >= 0 any(z) -{ 0 }-> 1 + (1 + s12) :|: s12 >= 0, s12 <= 2 * (z - 1), z - 1 >= 0 max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 max(z, z') -{ 0 }-> 1 + s10 :|: s10 >= 0, s10 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 min(z, z') -{ 0 }-> 1 + s22 :|: s22 >= 0, s22 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> 1 + s33 :|: s33 >= 0, s33 <= z - 1, s13 >= 0, s13 <= 2 * (z' - 2), z' - 2 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + s34 :|: s34 >= 0, s34 <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 1 + s35 :|: s35 >= 0, s35 <= z - 1, z' - 1 >= 0, z - 1 >= 0 Function symbols to be analyzed: {GCD} Previous analysis results are: ANY: runtime: O(n^1) [z], size: O(1) [0] MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] any: runtime: O(1) [0], size: O(n^1) [2*z] min: runtime: O(1) [0], size: O(n^1) [z'] MAX: runtime: O(n^1) [z'], size: O(1) [0] MINUS: runtime: EXP, size: O(1) [0] minus: runtime: O(1) [0], size: O(n^1) [z] GCD: runtime: INF, size: O(1) [0] ---------------------------------------- (109) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: any(v0) -> null_any [0] minus(v0, v1) -> null_minus [0] max(v0, v1) -> null_max [0] min(v0, v1) -> null_min [0] MIN(v0, v1) -> null_MIN [0] MAX(v0, v1) -> null_MAX [0] MINUS(v0, v1) -> null_MINUS [0] GCD(v0, v1) -> null_GCD [0] ANY(v0) -> null_ANY [0] And the following fresh constants: null_any, null_minus, null_max, null_min, null_MIN, null_MAX, null_MINUS, null_GCD, null_ANY ---------------------------------------- (110) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) [1] MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) [1] MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) [1] GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) [1] GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) [1] GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) [1] ANY(s(z0)) -> c11(ANY(z0)) [1] any(s(z0)) -> s(s(any(z0))) [0] any(z0) -> z0 [0] minus(z0, 0) -> z0 [0] minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) [0] max(z0, 0) -> z0 [0] max(0, z0) -> z0 [0] max(s(z0), s(z1)) -> s(max(z0, z1)) [0] min(z0, 0) -> 0 [0] min(0, z0) -> 0 [0] min(s(z0), s(z1)) -> s(min(z0, z1)) [0] any(v0) -> null_any [0] minus(v0, v1) -> null_minus [0] max(v0, v1) -> null_max [0] min(v0, v1) -> null_min [0] MIN(v0, v1) -> null_MIN [0] MAX(v0, v1) -> null_MAX [0] MINUS(v0, v1) -> null_MINUS [0] GCD(v0, v1) -> null_GCD [0] ANY(v0) -> null_ANY [0] The TRS has the following type information: MIN :: s:0:null_any:null_minus:null_max:null_min -> s:0:null_any:null_minus:null_max:null_min -> c2:null_MIN s :: s:0:null_any:null_minus:null_max:null_min -> s:0:null_any:null_minus:null_max:null_min c2 :: c2:null_MIN -> c2:null_MIN MAX :: s:0:null_any:null_minus:null_max:null_min -> s:0:null_any:null_minus:null_max:null_min -> c5:null_MAX c5 :: c5:null_MAX -> c5:null_MAX MINUS :: s:0:null_any:null_minus:null_max:null_min -> s:0:null_any:null_minus:null_max:null_min -> c7:null_MINUS c7 :: c7:null_MINUS -> c11:null_ANY -> c7:null_MINUS any :: s:0:null_any:null_minus:null_max:null_min -> s:0:null_any:null_minus:null_max:null_min ANY :: s:0:null_any:null_minus:null_max:null_min -> c11:null_ANY GCD :: s:0:null_any:null_minus:null_max:null_min -> s:0:null_any:null_minus:null_max:null_min -> c8:c9:c10:null_GCD c8 :: c8:c9:c10:null_GCD -> c7:null_MINUS -> c5:null_MAX -> c8:c9:c10:null_GCD minus :: s:0:null_any:null_minus:null_max:null_min -> s:0:null_any:null_minus:null_max:null_min -> s:0:null_any:null_minus:null_max:null_min max :: s:0:null_any:null_minus:null_max:null_min -> s:0:null_any:null_minus:null_max:null_min -> s:0:null_any:null_minus:null_max:null_min min :: s:0:null_any:null_minus:null_max:null_min -> s:0:null_any:null_minus:null_max:null_min -> s:0:null_any:null_minus:null_max:null_min c9 :: c8:c9:c10:null_GCD -> c7:null_MINUS -> c2:null_MIN -> c8:c9:c10:null_GCD c10 :: c8:c9:c10:null_GCD -> c2:null_MIN -> c8:c9:c10:null_GCD c11 :: c11:null_ANY -> c11:null_ANY 0 :: s:0:null_any:null_minus:null_max:null_min null_any :: s:0:null_any:null_minus:null_max:null_min null_minus :: s:0:null_any:null_minus:null_max:null_min null_max :: s:0:null_any:null_minus:null_max:null_min null_min :: s:0:null_any:null_minus:null_max:null_min null_MIN :: c2:null_MIN null_MAX :: c5:null_MAX null_MINUS :: c7:null_MINUS null_GCD :: c8:c9:c10:null_GCD null_ANY :: c11:null_ANY Rewrite Strategy: INNERMOST ---------------------------------------- (111) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 null_any => 0 null_minus => 0 null_max => 0 null_min => 0 null_MIN => 0 null_MAX => 0 null_MINUS => 0 null_GCD => 0 null_ANY => 0 ---------------------------------------- (112) Obligation: Complexity RNTS consisting of the following rules: ANY(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ANY(z) -{ 1 }-> 1 + ANY(z0) :|: z = 1 + z0, z0 >= 0 GCD(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z0, z1), min(z0, z1)), 1 + min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z0, z1), min(z0, z1)), 1 + min(z0, z1)) + MINUS(max(z0, z1), min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 GCD(z, z') -{ 1 }-> 1 + GCD(minus(max(z0, z1), min(z0, z1)), 1 + min(z0, z1)) + MINUS(max(z0, z1), min(z0, z1)) + MAX(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 MAX(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 MAX(z, z') -{ 1 }-> 1 + MAX(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 MIN(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 MIN(z, z') -{ 1 }-> 1 + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 MINUS(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 MINUS(z, z') -{ 1 }-> 1 + MINUS(z0, any(z1)) + ANY(z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 any(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0 any(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 any(z) -{ 0 }-> 1 + (1 + any(z0)) :|: z = 1 + z0, z0 >= 0 max(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0 max(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0 max(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 max(z, z') -{ 0 }-> 1 + max(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 min(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0 min(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0 min(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 min(z, z') -{ 0 }-> 1 + min(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 minus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 minus(z, z') -{ 0 }-> 1 + minus(z0, any(z1)) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINUS(s(z0), s(z1)) -> c7(MINUS(z0, any(z1)), ANY(z1)) by MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(any(z0)))), ANY(s(z0))) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(any(z0)))), ANY(s(z0))) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(any(z0)))), ANY(s(z0))) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, GCD_2, ANY_1, MINUS_2 Compound Symbols: c2_1, c5_1, c8_3, c9_3, c10_2, c11_1, c7_2 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), s(z1)) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MAX(z0, z1)) by GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0)), MAX(z0, 0)) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0)), MAX(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0)), MAX(z0, 0)) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0)), MAX(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0)), MAX(z0, 0)) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0)), MAX(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(any(z0)))), ANY(s(z0))) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0)), MAX(z0, 0)) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0)), MAX(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0)), MAX(z0, 0)) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0)), MAX(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0)), MAX(z0, 0)) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0)), MAX(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(any(z0)))), ANY(s(z0))) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0)), MAX(z0, 0)) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0)), MAX(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0)), MAX(z0, 0)) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0)), MAX(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0)), MAX(z0, 0)) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0)), MAX(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, GCD_2, ANY_1, MINUS_2 Compound Symbols: c2_1, c5_1, c9_3, c10_2, c11_1, c7_2, c8_3 ---------------------------------------- (117) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(any(z0)))), ANY(s(z0))) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(any(z0)))), ANY(s(z0))) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, GCD_2, ANY_1, MINUS_2 Compound Symbols: c2_1, c5_1, c9_3, c10_2, c11_1, c7_2, c8_3, c8_2 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), s(z1)) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MINUS(max(z0, z1), min(z0, z1)), MIN(z0, z1)) by GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0)), MIN(z0, 0)) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0)), MIN(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0)), MIN(z0, 0)) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0)), MIN(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0)), MIN(z0, 0)) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0)), MIN(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(any(z0)))), ANY(s(z0))) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0)), MIN(z0, 0)) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0)), MIN(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0)), MIN(z0, 0)) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0)), MIN(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0)), MIN(z0, 0)) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0)), MIN(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(any(z0)))), ANY(s(z0))) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0)), MIN(z0, 0)) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0)), MIN(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0)), MIN(z0, 0)) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0)), MIN(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0)), MIN(z0, 0)) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0)), MIN(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, GCD_2, ANY_1, MINUS_2 Compound Symbols: c2_1, c5_1, c10_2, c11_1, c7_2, c8_3, c8_2, c9_3 ---------------------------------------- (121) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(any(z0)))), ANY(s(z0))) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(any(z0)))), ANY(s(z0))) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, GCD_2, ANY_1, MINUS_2 Compound Symbols: c2_1, c5_1, c10_2, c11_1, c7_2, c8_3, c8_2, c9_3, c9_2 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), s(z1)) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1))), MIN(z0, z1)) by GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MIN(z0, 0)) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0)), MIN(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MIN(z0, 0)) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0))), MIN(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MIN(z0, 0)) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MIN(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(any(z0)))), ANY(s(z0))) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MIN(z0, 0)) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0)), MIN(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MIN(z0, 0)) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0))), MIN(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MIN(z0, 0)) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MIN(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(any(z0)))), ANY(s(z0))) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MIN(z0, 0)) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0)), MIN(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MIN(z0, 0)) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0))), MIN(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MIN(z0, 0)) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MIN(0, z0)) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c8_3, c8_2, c9_3, c9_2, c10_2 ---------------------------------------- (125) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(any(z0)))), ANY(s(z0))) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(any(z0)))), ANY(s(z0))) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(any(z0)))), ANY(s(z0))) by MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1 ---------------------------------------- (129) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) by GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) by GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) by GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c8_2, c9_3, c9_2, c10_2, c10_1, c8_3 ---------------------------------------- (135) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) by GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c8_2, c9_3, c9_2, c10_2, c10_1, c8_3 ---------------------------------------- (137) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) by GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c8_2, c9_3, c9_2, c10_2, c10_1, c8_3 ---------------------------------------- (139) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) by GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c8_2, c9_3, c9_2, c10_2, c10_1, c8_3 ---------------------------------------- (141) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) by GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c8_2, c9_3, c9_2, c10_2, c10_1, c8_3 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) by GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c8_2, c9_3, c9_2, c10_2, c10_1, c8_3 ---------------------------------------- (145) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) by GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c9_3, c9_2, c10_2, c10_1, c8_3, c8_2 ---------------------------------------- (147) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c9_3, c9_2, c10_2, c10_1, c8_3, c8_2 ---------------------------------------- (149) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c9_3, c9_2, c10_2, c10_1, c8_3, c8_2 ---------------------------------------- (151) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c9_2, c10_2, c10_1, c8_3, c8_2, c9_3 ---------------------------------------- (153) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) by GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c9_2, c10_2, c10_1, c8_3, c8_2, c9_3 ---------------------------------------- (155) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) by GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c9_2, c10_2, c10_1, c8_3, c8_2, c9_3 ---------------------------------------- (157) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) by GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c9_2, c10_2, c10_1, c8_3, c8_2, c9_3 ---------------------------------------- (159) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) by GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c9_2, c10_2, c10_1, c8_3, c8_2, c9_3 ---------------------------------------- (161) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) by GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c9_2, c10_2, c10_1, c8_3, c8_2, c9_3 ---------------------------------------- (163) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) by GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c10_2, c10_1, c8_3, c8_2, c9_3, c9_2 ---------------------------------------- (165) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) by GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c10_2, c10_1, c8_3, c8_2, c9_3, c9_2 ---------------------------------------- (167) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) by GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c10_2, c10_1, c8_3, c8_2, c9_3, c9_2 ---------------------------------------- (169) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) by GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c10_1, c8_3, c8_2, c9_3, c9_2, c10_2 ---------------------------------------- (171) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), min(z0, 0)), s(0))) by GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c10_1, c8_3, c8_2, c9_3, c9_2, c10_2 ---------------------------------------- (173) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), min(0, z0)), s(0))) by GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c10_1, c8_3, c8_2, c9_3, c9_2, c10_2 ---------------------------------------- (175) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(min(z0, 0)))) by GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c10_1, c8_3, c8_2, c9_3, c9_2, c10_2 ---------------------------------------- (177) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(min(0, z0)))) by GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c10_1, c8_3, c8_2, c9_3, c9_2, c10_2 ---------------------------------------- (179) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(min(z0, 0)))) by GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c10_1, c8_3, c8_2, c9_3, c9_2, c10_2 ---------------------------------------- (181) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(min(0, z0)))) by GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, ANY_1, MINUS_2, GCD_2 Compound Symbols: c2_1, c5_1, c11_1, c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1 ---------------------------------------- (183) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) by MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MIN(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MIN(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MIN(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MAX_2, ANY_1, MINUS_2, GCD_2, MIN_2 Compound Symbols: c5_1, c11_1, c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1, c2_1 ---------------------------------------- (185) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing tuple parts ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MAX_2, ANY_1, MINUS_2, GCD_2, MIN_2 Compound Symbols: c5_1, c11_1, c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1, c2_1 ---------------------------------------- (187) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) by MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) S tuples: ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: ANY_1, MINUS_2, GCD_2, MIN_2, MAX_2 Compound Symbols: c11_1, c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1, c2_1, c5_1 ---------------------------------------- (189) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) S tuples: ANY(s(z0)) -> c11(ANY(z0)) MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: ANY_1, MINUS_2, GCD_2, MIN_2, MAX_2 Compound Symbols: c11_1, c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1, c2_1, c5_1 ---------------------------------------- (191) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ANY(s(z0)) -> c11(ANY(z0)) by ANY(s(s(y0))) -> c11(ANY(s(y0))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) S tuples: MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MINUS_2, GCD_2, MIN_2, MAX_2, ANY_1 Compound Symbols: c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1, c2_1, c5_1, c11_1 ---------------------------------------- (193) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MINUS(s(x0), s(z0)) -> c7(MINUS(x0, z0), ANY(z0)) by MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) S tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MINUS_2, GCD_2, MIN_2, MAX_2, ANY_1 Compound Symbols: c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1, c2_1, c5_1, c11_1 ---------------------------------------- (195) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MINUS(s(x0), s(s(z0))) -> c7(MINUS(x0, s(s(z0))), ANY(s(z0))) by MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) S tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MINUS_2, GCD_2, MIN_2, MAX_2, ANY_1 Compound Symbols: c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1, c2_1, c5_1, c11_1 ---------------------------------------- (197) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) by GCD(s(z0), s(0)) -> c8(GCD(z0, s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) GCD(s(z0), s(0)) -> c8(GCD(z0, s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) S tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) GCD(s(z0), s(0)) -> c8(GCD(z0, s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MINUS_2, GCD_2, MIN_2, MAX_2, ANY_1 Compound Symbols: c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1, c2_1, c5_1, c11_1 ---------------------------------------- (199) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) by GCD(s(0), s(0)) -> c8(GCD(0, s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) GCD(s(z0), s(0)) -> c8(GCD(z0, s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(0, s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) S tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) GCD(s(z0), s(0)) -> c8(GCD(z0, s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(0, s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MINUS_2, GCD_2, MIN_2, MAX_2, ANY_1 Compound Symbols: c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1, c2_1, c5_1, c11_1 ---------------------------------------- (201) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) GCD(s(z0), s(0)) -> c8(GCD(z0, s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(MINUS(max(0, 0), min(0, 0))) S tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) GCD(s(z0), s(0)) -> c8(GCD(z0, s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(MINUS(max(0, 0), min(0, 0))) K tuples:none Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MINUS_2, GCD_2, MIN_2, MAX_2, ANY_1 Compound Symbols: c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1, c2_1, c5_1, c11_1, c8_1 ---------------------------------------- (203) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. GCD(s(0), s(0)) -> c8(MINUS(max(0, 0), min(0, 0))) We considered the (Usable) Rules:none And the Tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) GCD(s(z0), s(0)) -> c8(GCD(z0, s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(MINUS(max(0, 0), min(0, 0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ANY(x_1)) = x_1 POL(GCD(x_1, x_2)) = [1] POL(MAX(x_1, x_2)) = x_1 + x_2 POL(MIN(x_1, x_2)) = x_1 + x_2 POL(MINUS(x_1, x_2)) = 0 POL(any(x_1)) = [1] + x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(max(x_1, x_2)) = 0 POL(min(x_1, x_2)) = 0 POL(minus(x_1, x_2)) = [1] POL(s(x_1)) = 0 ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) GCD(s(z0), s(0)) -> c8(GCD(z0, s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(MINUS(max(0, 0), min(0, 0))) S tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) GCD(s(z0), s(0)) -> c8(GCD(z0, s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) K tuples: GCD(s(0), s(0)) -> c8(MINUS(max(0, 0), min(0, 0))) Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MINUS_2, GCD_2, MIN_2, MAX_2, ANY_1 Compound Symbols: c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1, c2_1, c5_1, c11_1, c8_1 ---------------------------------------- (205) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) by GCD(s(0), s(0)) -> c8(GCD(0, s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) GCD(s(z0), s(0)) -> c8(GCD(z0, s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(0)) -> c8(GCD(0, s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) S tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) GCD(s(z0), s(0)) -> c8(GCD(z0, s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) K tuples: GCD(s(0), s(0)) -> c8(MINUS(max(0, 0), min(0, 0))) Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MINUS_2, GCD_2, MIN_2, MAX_2, ANY_1 Compound Symbols: c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1, c2_1, c5_1, c11_1, c8_1 ---------------------------------------- (207) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) GCD(s(z0), s(0)) -> c8(GCD(z0, s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(MINUS(max(0, 0), min(0, 0))) S tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) GCD(s(z0), s(0)) -> c8(GCD(z0, s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) K tuples: GCD(s(0), s(0)) -> c8(MINUS(max(0, 0), min(0, 0))) Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MINUS_2, GCD_2, MIN_2, MAX_2, ANY_1 Compound Symbols: c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1, c2_1, c5_1, c11_1, c8_1 ---------------------------------------- (209) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) by MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) GCD(s(z0), s(0)) -> c8(GCD(z0, s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(MINUS(max(0, 0), min(0, 0))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) S tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) GCD(s(z0), s(0)) -> c8(GCD(z0, s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) K tuples: GCD(s(0), s(0)) -> c8(MINUS(max(0, 0), min(0, 0))) Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MINUS_2, GCD_2, MAX_2, ANY_1, MIN_2 Compound Symbols: c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1, c5_1, c11_1, c8_1, c2_1 ---------------------------------------- (211) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) by GCD(s(0), s(z0)) -> c8(GCD(z0, s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: any(s(z0)) -> s(s(any(z0))) any(z0) -> z0 minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> s(minus(z0, any(z1))) max(z0, 0) -> z0 max(0, z0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) Tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) GCD(s(z0), s(0)) -> c8(GCD(z0, s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(MINUS(max(0, 0), min(0, 0))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) GCD(s(0), s(z0)) -> c8(GCD(z0, s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) S tuples: MINUS(s(x0), s(s(s(z0)))) -> c7(MINUS(x0, s(s(s(s(any(z0)))))), ANY(s(s(z0)))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c8(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c8(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c8(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c8(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c8(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c8(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c8(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c8(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(0), s(x0)) -> c8(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c9(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MINUS(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c9(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MINUS(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c9(GCD(minus(max(z0, 0), 0), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(max(0, 0), 0), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, min(z0, 0)), s(0)), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, min(0, 0)), s(0)), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(z0)) -> c9(GCD(minus(max(0, z0), 0), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, min(0, z0)), s(0)), MINUS(max(0, z0), min(0, z0))) GCD(s(x0), s(0)) -> c9(GCD(max(x0, 0), s(min(x0, 0))), MINUS(max(x0, 0), min(x0, 0))) GCD(s(z0), s(0)) -> c9(GCD(minus(z0, 0), s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) GCD(s(0), s(0)) -> c9(GCD(minus(0, 0), s(min(0, 0))), MINUS(max(0, 0), min(0, 0))) GCD(s(0), s(x0)) -> c9(GCD(max(0, x0), s(min(0, x0))), MINUS(max(0, x0), min(0, x0))) GCD(s(0), s(z0)) -> c9(GCD(minus(z0, 0), s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), s(s(s(min(z0, z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(max(s(z0), s(z1)), s(min(z0, z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(s(z0)), s(s(z1))) -> c10(GCD(minus(s(max(z0, z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), MIN(s(z0), s(z1))) GCD(s(s(s(z0))), s(s(s(z1)))) -> c10(GCD(minus(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), s(min(s(s(z0)), s(s(z1))))), MIN(s(s(z0)), s(s(z1)))) GCD(s(z0), s(0)) -> c10(GCD(minus(max(z0, 0), 0), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(max(0, 0), 0), s(0))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, min(z0, 0)), s(0))) GCD(s(0), s(0)) -> c10(GCD(minus(0, min(0, 0)), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(max(0, z0), 0), s(0))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, min(0, z0)), s(0))) GCD(s(x0), s(0)) -> c10(GCD(max(x0, 0), s(min(x0, 0)))) GCD(s(z0), s(0)) -> c10(GCD(minus(z0, 0), s(min(z0, 0)))) GCD(s(0), s(0)) -> c10(GCD(minus(0, 0), s(min(0, 0)))) GCD(s(0), s(x0)) -> c10(GCD(max(0, x0), s(min(0, x0)))) GCD(s(0), s(z0)) -> c10(GCD(minus(z0, 0), s(min(0, z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c9(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c9(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0)))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0)))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0))))) GCD(s(s(z0)), s(s(0))) -> c10(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0))))) GCD(s(s(0)), s(s(z0))) -> c10(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0))))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), min(s(z0), s(0))), s(s(0))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), min(s(0), s(z0))), s(s(0))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(max(s(z0), s(0)), s(0)), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(max(s(0), s(z0)), s(0)), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) GCD(s(s(z0)), s(s(0))) -> c8(GCD(minus(s(z0), min(s(z0), s(0))), s(min(s(z0), s(0)))), MINUS(max(s(z0), s(0)), min(s(z0), s(0)))) GCD(s(s(0)), s(s(z0))) -> c8(GCD(minus(s(z0), min(s(0), s(z0))), s(min(s(0), s(z0)))), MINUS(max(s(0), s(z0)), min(s(0), s(z0)))) ANY(s(s(y0))) -> c11(ANY(s(y0))) MINUS(s(s(y0)), s(s(y1))) -> c7(MINUS(s(y0), s(y1)), ANY(s(y1))) MINUS(s(s(y0)), s(s(s(s(y1))))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(s(y1))))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(y1))), ANY(s(s(y1)))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(y0))), ANY(s(s(y0)))) MINUS(s(s(y0)), s(s(s(y1)))) -> c7(MINUS(s(y0), s(s(s(y1)))), ANY(s(s(y1)))) MINUS(s(s(y0)), s(s(z1))) -> c7(MINUS(s(y0), s(s(z1))), ANY(s(z1))) MINUS(s(z0), s(s(s(y0)))) -> c7(MINUS(z0, s(s(s(y0)))), ANY(s(s(y0)))) MINUS(s(s(s(y0))), s(s(z1))) -> c7(MINUS(s(s(y0)), s(s(z1))), ANY(s(z1))) MINUS(s(s(s(y0))), s(s(s(s(y1))))) -> c7(MINUS(s(s(y0)), s(s(s(s(y1))))), ANY(s(s(s(y1))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(s(y1)))), ANY(s(s(y1)))) GCD(s(z0), s(0)) -> c8(GCD(z0, s(min(z0, 0))), MINUS(max(z0, 0), min(z0, 0))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) GCD(s(0), s(z0)) -> c8(GCD(z0, s(min(0, z0))), MINUS(max(0, z0), min(0, z0))) K tuples: GCD(s(0), s(0)) -> c8(MINUS(max(0, 0), min(0, 0))) Defined Rule Symbols: any_1, minus_2, max_2, min_2 Defined Pair Symbols: MINUS_2, GCD_2, MAX_2, ANY_1, MIN_2 Compound Symbols: c7_2, c8_3, c8_2, c9_3, c9_2, c10_2, c10_1, c5_1, c11_1, c8_1, c2_1 ---------------------------------------- (213) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (214) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: min(x, 0) -> 0 [1] min(0, y) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] max(x, 0) -> x [1] max(0, y) -> y [1] max(s(x), s(y)) -> s(max(x, y)) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> s(minus(x, any(y))) [1] gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y))) [1] any(s(x)) -> s(s(any(x))) [1] any(x) -> x [1] Rewrite Strategy: INNERMOST ---------------------------------------- (215) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (216) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: min(x, 0) -> 0 [1] min(0, y) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] max(x, 0) -> x [1] max(0, y) -> y [1] max(s(x), s(y)) -> s(max(x, y)) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> s(minus(x, any(y))) [1] gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y))) [1] any(s(x)) -> s(s(any(x))) [1] any(x) -> x [1] The TRS has the following type information: min :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s max :: 0:s -> 0:s -> 0:s minus :: 0:s -> 0:s -> 0:s any :: 0:s -> 0:s gcd :: 0:s -> 0:s -> gcd Rewrite Strategy: INNERMOST ---------------------------------------- (217) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: gcd_2 (c) The following functions are completely defined: any_1 minus_2 max_2 min_2 Due to the following rules being added: minus(v0, v1) -> 0 [0] And the following fresh constants: const ---------------------------------------- (218) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: min(x, 0) -> 0 [1] min(0, y) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] max(x, 0) -> x [1] max(0, y) -> y [1] max(s(x), s(y)) -> s(max(x, y)) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> s(minus(x, any(y))) [1] gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y))) [1] any(s(x)) -> s(s(any(x))) [1] any(x) -> x [1] minus(v0, v1) -> 0 [0] The TRS has the following type information: min :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s max :: 0:s -> 0:s -> 0:s minus :: 0:s -> 0:s -> 0:s any :: 0:s -> 0:s gcd :: 0:s -> 0:s -> gcd const :: gcd Rewrite Strategy: INNERMOST ---------------------------------------- (219) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (220) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: min(x, 0) -> 0 [1] min(0, y) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] max(x, 0) -> x [1] max(0, y) -> y [1] max(s(x), s(y)) -> s(max(x, y)) [1] minus(x, 0) -> x [1] minus(s(x), s(s(x'))) -> s(minus(x, s(s(any(x'))))) [2] minus(s(x), s(y)) -> s(minus(x, y)) [2] gcd(s(x), s(0)) -> gcd(minus(x, 0), s(0)) [4] gcd(s(0), s(0)) -> gcd(minus(0, 0), s(0)) [4] gcd(s(0), s(0)) -> gcd(minus(0, 0), s(0)) [4] gcd(s(0), s(0)) -> gcd(minus(0, 0), s(0)) [4] gcd(s(0), s(0)) -> gcd(minus(0, 0), s(0)) [4] gcd(s(0), s(0)) -> gcd(minus(0, 0), s(0)) [4] gcd(s(0), s(0)) -> gcd(minus(0, 0), s(0)) [4] gcd(s(0), s(y)) -> gcd(minus(y, 0), s(0)) [4] gcd(s(s(x'')), s(s(y'))) -> gcd(minus(s(max(x'', y')), s(min(x'', y'))), s(s(min(x'', y')))) [4] any(s(x)) -> s(s(any(x))) [1] any(x) -> x [1] minus(v0, v1) -> 0 [0] The TRS has the following type information: min :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s max :: 0:s -> 0:s -> 0:s minus :: 0:s -> 0:s -> 0:s any :: 0:s -> 0:s gcd :: 0:s -> 0:s -> gcd const :: gcd Rewrite Strategy: INNERMOST ---------------------------------------- (221) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 const => 0 ---------------------------------------- (222) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> x :|: x >= 0, z = x any(z) -{ 1 }-> 1 + (1 + any(x)) :|: x >= 0, z = 1 + x gcd(z, z') -{ 4 }-> gcd(minus(x, 0), 1 + 0) :|: x >= 0, z' = 1 + 0, z = 1 + x gcd(z, z') -{ 4 }-> gcd(minus(y, 0), 1 + 0) :|: z' = 1 + y, z = 1 + 0, y >= 0 gcd(z, z') -{ 4 }-> gcd(minus(0, 0), 1 + 0) :|: z = 1 + 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(1 + max(x'', y'), 1 + min(x'', y')), 1 + (1 + min(x'', y'))) :|: z' = 1 + (1 + y'), y' >= 0, x'' >= 0, z = 1 + (1 + x'') max(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 max(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y max(z, z') -{ 1 }-> 1 + max(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x min(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 min(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y min(z, z') -{ 1 }-> 1 + min(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 minus(z, z') -{ 2 }-> 1 + minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 2 }-> 1 + minus(x, 1 + (1 + any(x'))) :|: z' = 1 + (1 + x'), x >= 0, x' >= 0, z = 1 + x ---------------------------------------- (223) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (224) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> z :|: z >= 0 any(z) -{ 1 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(0, 0), 1 + 0) :|: z = 1 + 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z - 1, 0), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0) :|: z = 1 + 0, z' - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(1 + max(z - 2, z' - 2), 1 + min(z - 2, z' - 2)), 1 + (1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z - 1 >= 0, z' - 2 >= 0 ---------------------------------------- (225) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { any } { max } { min } { minus } { gcd } ---------------------------------------- (226) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> z :|: z >= 0 any(z) -{ 1 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(0, 0), 1 + 0) :|: z = 1 + 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z - 1, 0), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0) :|: z = 1 + 0, z' - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(1 + max(z - 2, z' - 2), 1 + min(z - 2, z' - 2)), 1 + (1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z - 1 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {any}, {max}, {min}, {minus}, {gcd} ---------------------------------------- (227) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (228) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> z :|: z >= 0 any(z) -{ 1 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(0, 0), 1 + 0) :|: z = 1 + 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z - 1, 0), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0) :|: z = 1 + 0, z' - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(1 + max(z - 2, z' - 2), 1 + min(z - 2, z' - 2)), 1 + (1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z - 1 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {any}, {max}, {min}, {minus}, {gcd} ---------------------------------------- (229) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: any after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z ---------------------------------------- (230) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> z :|: z >= 0 any(z) -{ 1 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(0, 0), 1 + 0) :|: z = 1 + 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z - 1, 0), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0) :|: z = 1 + 0, z' - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(1 + max(z - 2, z' - 2), 1 + min(z - 2, z' - 2)), 1 + (1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z - 1 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {any}, {max}, {min}, {minus}, {gcd} Previous analysis results are: any: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (231) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: any after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (232) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> z :|: z >= 0 any(z) -{ 1 }-> 1 + (1 + any(z - 1)) :|: z - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(0, 0), 1 + 0) :|: z = 1 + 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z - 1, 0), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0) :|: z = 1 + 0, z' - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(1 + max(z - 2, z' - 2), 1 + min(z - 2, z' - 2)), 1 + (1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, 1 + (1 + any(z' - 2))) :|: z - 1 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {max}, {min}, {minus}, {gcd} Previous analysis results are: any: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (233) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (234) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> z :|: z >= 0 any(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(0, 0), 1 + 0) :|: z = 1 + 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z - 1, 0), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0) :|: z = 1 + 0, z' - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(1 + max(z - 2, z' - 2), 1 + min(z - 2, z' - 2)), 1 + (1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> 1 + minus(z - 1, 1 + (1 + s)) :|: s >= 0, s <= 2 * (z' - 2), z - 1 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {max}, {min}, {minus}, {gcd} Previous analysis results are: any: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (235) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: max after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (236) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> z :|: z >= 0 any(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(0, 0), 1 + 0) :|: z = 1 + 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z - 1, 0), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0) :|: z = 1 + 0, z' - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(1 + max(z - 2, z' - 2), 1 + min(z - 2, z' - 2)), 1 + (1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> 1 + minus(z - 1, 1 + (1 + s)) :|: s >= 0, s <= 2 * (z' - 2), z - 1 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {max}, {min}, {minus}, {gcd} Previous analysis results are: any: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (237) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: max after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (238) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> z :|: z >= 0 any(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(0, 0), 1 + 0) :|: z = 1 + 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z - 1, 0), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0) :|: z = 1 + 0, z' - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(1 + max(z - 2, z' - 2), 1 + min(z - 2, z' - 2)), 1 + (1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> 1 + minus(z - 1, 1 + (1 + s)) :|: s >= 0, s <= 2 * (z' - 2), z - 1 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {min}, {minus}, {gcd} Previous analysis results are: any: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] ---------------------------------------- (239) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (240) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> z :|: z >= 0 any(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(0, 0), 1 + 0) :|: z = 1 + 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z - 1, 0), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0) :|: z = 1 + 0, z' - 1 >= 0 gcd(z, z') -{ 4 + z' }-> gcd(minus(1 + s1, 1 + min(z - 2, z' - 2)), 1 + (1 + min(z - 2, z' - 2))) :|: s1 >= 0, s1 <= z' - 2 + (z - 2), z' - 2 >= 0, z - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> 1 + minus(z - 1, 1 + (1 + s)) :|: s >= 0, s <= 2 * (z' - 2), z - 1 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {min}, {minus}, {gcd} Previous analysis results are: any: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] ---------------------------------------- (241) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: min after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (242) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> z :|: z >= 0 any(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(0, 0), 1 + 0) :|: z = 1 + 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z - 1, 0), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0) :|: z = 1 + 0, z' - 1 >= 0 gcd(z, z') -{ 4 + z' }-> gcd(minus(1 + s1, 1 + min(z - 2, z' - 2)), 1 + (1 + min(z - 2, z' - 2))) :|: s1 >= 0, s1 <= z' - 2 + (z - 2), z' - 2 >= 0, z - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> 1 + minus(z - 1, 1 + (1 + s)) :|: s >= 0, s <= 2 * (z' - 2), z - 1 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {min}, {minus}, {gcd} Previous analysis results are: any: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] min: runtime: ?, size: O(n^1) [z] ---------------------------------------- (243) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: min after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (244) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> z :|: z >= 0 any(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(0, 0), 1 + 0) :|: z = 1 + 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z - 1, 0), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0) :|: z = 1 + 0, z' - 1 >= 0 gcd(z, z') -{ 4 + z' }-> gcd(minus(1 + s1, 1 + min(z - 2, z' - 2)), 1 + (1 + min(z - 2, z' - 2))) :|: s1 >= 0, s1 <= z' - 2 + (z - 2), z' - 2 >= 0, z - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> 1 + minus(z - 1, 1 + (1 + s)) :|: s >= 0, s <= 2 * (z' - 2), z - 1 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {minus}, {gcd} Previous analysis results are: any: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] ---------------------------------------- (245) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (246) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> z :|: z >= 0 any(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(0, 0), 1 + 0) :|: z = 1 + 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z - 1, 0), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0) :|: z = 1 + 0, z' - 1 >= 0 gcd(z, z') -{ 4 + 3*z' }-> gcd(minus(1 + s1, 1 + s3), 1 + (1 + s4)) :|: s3 >= 0, s3 <= z - 2, s4 >= 0, s4 <= z - 2, s1 >= 0, s1 <= z' - 2 + (z - 2), z' - 2 >= 0, z - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 2 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> 1 + minus(z - 1, 1 + (1 + s)) :|: s >= 0, s <= 2 * (z' - 2), z - 1 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {minus}, {gcd} Previous analysis results are: any: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] ---------------------------------------- (247) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (248) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> z :|: z >= 0 any(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(0, 0), 1 + 0) :|: z = 1 + 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z - 1, 0), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0) :|: z = 1 + 0, z' - 1 >= 0 gcd(z, z') -{ 4 + 3*z' }-> gcd(minus(1 + s1, 1 + s3), 1 + (1 + s4)) :|: s3 >= 0, s3 <= z - 2, s4 >= 0, s4 <= z - 2, s1 >= 0, s1 <= z' - 2 + (z - 2), z' - 2 >= 0, z - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 2 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> 1 + minus(z - 1, 1 + (1 + s)) :|: s >= 0, s <= 2 * (z' - 2), z - 1 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {minus}, {gcd} Previous analysis results are: any: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] minus: runtime: ?, size: O(n^1) [z] ---------------------------------------- (249) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: minus after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (250) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> z :|: z >= 0 any(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 gcd(z, z') -{ 4 }-> gcd(minus(0, 0), 1 + 0) :|: z = 1 + 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z - 1, 0), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0) :|: z = 1 + 0, z' - 1 >= 0 gcd(z, z') -{ 4 + 3*z' }-> gcd(minus(1 + s1, 1 + s3), 1 + (1 + s4)) :|: s3 >= 0, s3 <= z - 2, s4 >= 0, s4 <= z - 2, s1 >= 0, s1 <= z' - 2 + (z - 2), z' - 2 >= 0, z - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 2 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> 1 + minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> 1 + minus(z - 1, 1 + (1 + s)) :|: s >= 0, s <= 2 * (z' - 2), z - 1 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {gcd} Previous analysis results are: any: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] minus: runtime: EXP, size: O(n^1) [z] ---------------------------------------- (251) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (252) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> z :|: z >= 0 any(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 gcd(z, z') -{ 4 + inf3 + 3*z' }-> gcd(s10, 1 + (1 + s4)) :|: s10 >= 0, s10 <= 1 + s1, s3 >= 0, s3 <= z - 2, s4 >= 0, s4 <= z - 2, s1 >= 0, s1 <= z' - 2 + (z - 2), z' - 2 >= 0, z - 2 >= 0 gcd(z, z') -{ 4 + inf'' }-> gcd(s7, 1 + 0) :|: s7 >= 0, s7 <= z - 1, z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 4 + inf1 }-> gcd(s8, 1 + 0) :|: s8 >= 0, s8 <= 0, z = 1 + 0, z' = 1 + 0 gcd(z, z') -{ 4 + inf2 }-> gcd(s9, 1 + 0) :|: s9 >= 0, s9 <= z' - 1, z = 1 + 0, z' - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 2 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 1 + inf + z' }-> 1 + s5 :|: s5 >= 0, s5 <= z - 1, s >= 0, s <= 2 * (z' - 2), z - 1 >= 0, z' - 2 >= 0 minus(z, z') -{ 2 + inf' }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0 Function symbols to be analyzed: {gcd} Previous analysis results are: any: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] minus: runtime: EXP, size: O(n^1) [z] ---------------------------------------- (253) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: gcd after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (254) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> z :|: z >= 0 any(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 gcd(z, z') -{ 4 + inf3 + 3*z' }-> gcd(s10, 1 + (1 + s4)) :|: s10 >= 0, s10 <= 1 + s1, s3 >= 0, s3 <= z - 2, s4 >= 0, s4 <= z - 2, s1 >= 0, s1 <= z' - 2 + (z - 2), z' - 2 >= 0, z - 2 >= 0 gcd(z, z') -{ 4 + inf'' }-> gcd(s7, 1 + 0) :|: s7 >= 0, s7 <= z - 1, z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 4 + inf1 }-> gcd(s8, 1 + 0) :|: s8 >= 0, s8 <= 0, z = 1 + 0, z' = 1 + 0 gcd(z, z') -{ 4 + inf2 }-> gcd(s9, 1 + 0) :|: s9 >= 0, s9 <= z' - 1, z = 1 + 0, z' - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 2 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 1 + inf + z' }-> 1 + s5 :|: s5 >= 0, s5 <= z - 1, s >= 0, s <= 2 * (z' - 2), z - 1 >= 0, z' - 2 >= 0 minus(z, z') -{ 2 + inf' }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0 Function symbols to be analyzed: {gcd} Previous analysis results are: any: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] minus: runtime: EXP, size: O(n^1) [z] gcd: runtime: ?, size: O(1) [0] ---------------------------------------- (255) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: gcd after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (256) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> z :|: z >= 0 any(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 gcd(z, z') -{ 4 + inf3 + 3*z' }-> gcd(s10, 1 + (1 + s4)) :|: s10 >= 0, s10 <= 1 + s1, s3 >= 0, s3 <= z - 2, s4 >= 0, s4 <= z - 2, s1 >= 0, s1 <= z' - 2 + (z - 2), z' - 2 >= 0, z - 2 >= 0 gcd(z, z') -{ 4 + inf'' }-> gcd(s7, 1 + 0) :|: s7 >= 0, s7 <= z - 1, z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 4 + inf1 }-> gcd(s8, 1 + 0) :|: s8 >= 0, s8 <= 0, z = 1 + 0, z' = 1 + 0 gcd(z, z') -{ 4 + inf2 }-> gcd(s9, 1 + 0) :|: s9 >= 0, s9 <= z' - 1, z = 1 + 0, z' - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 2 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 1 + inf + z' }-> 1 + s5 :|: s5 >= 0, s5 <= z - 1, s >= 0, s <= 2 * (z' - 2), z - 1 >= 0, z' - 2 >= 0 minus(z, z') -{ 2 + inf' }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0 Function symbols to be analyzed: {gcd} Previous analysis results are: any: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] minus: runtime: EXP, size: O(n^1) [z] gcd: runtime: INF, size: O(1) [0] ---------------------------------------- (257) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: minus(v0, v1) -> null_minus [0] gcd(v0, v1) -> null_gcd [0] min(v0, v1) -> null_min [0] max(v0, v1) -> null_max [0] And the following fresh constants: null_minus, null_gcd, null_min, null_max ---------------------------------------- (258) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: min(x, 0) -> 0 [1] min(0, y) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] max(x, 0) -> x [1] max(0, y) -> y [1] max(s(x), s(y)) -> s(max(x, y)) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> s(minus(x, any(y))) [1] gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y))) [1] any(s(x)) -> s(s(any(x))) [1] any(x) -> x [1] minus(v0, v1) -> null_minus [0] gcd(v0, v1) -> null_gcd [0] min(v0, v1) -> null_min [0] max(v0, v1) -> null_max [0] The TRS has the following type information: min :: 0:s:null_minus:null_min:null_max -> 0:s:null_minus:null_min:null_max -> 0:s:null_minus:null_min:null_max 0 :: 0:s:null_minus:null_min:null_max s :: 0:s:null_minus:null_min:null_max -> 0:s:null_minus:null_min:null_max max :: 0:s:null_minus:null_min:null_max -> 0:s:null_minus:null_min:null_max -> 0:s:null_minus:null_min:null_max minus :: 0:s:null_minus:null_min:null_max -> 0:s:null_minus:null_min:null_max -> 0:s:null_minus:null_min:null_max any :: 0:s:null_minus:null_min:null_max -> 0:s:null_minus:null_min:null_max gcd :: 0:s:null_minus:null_min:null_max -> 0:s:null_minus:null_min:null_max -> null_gcd null_minus :: 0:s:null_minus:null_min:null_max null_gcd :: null_gcd null_min :: 0:s:null_minus:null_min:null_max null_max :: 0:s:null_minus:null_min:null_max Rewrite Strategy: INNERMOST ---------------------------------------- (259) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 null_minus => 0 null_gcd => 0 null_min => 0 null_max => 0 ---------------------------------------- (260) Obligation: Complexity RNTS consisting of the following rules: any(z) -{ 1 }-> x :|: x >= 0, z = x any(z) -{ 1 }-> 1 + (1 + any(x)) :|: x >= 0, z = 1 + x gcd(z, z') -{ 1 }-> gcd(minus(max(x, y), min(x, y)), 1 + min(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x gcd(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 max(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 max(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y max(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 max(z, z') -{ 1 }-> 1 + max(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x min(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 min(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y min(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 min(z, z') -{ 1 }-> 1 + min(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 minus(z, z') -{ 1 }-> 1 + minus(x, any(y)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x Only complete derivations are relevant for the runtime complexity.