KILLED proof of input_K9OoESCKPC.trs # AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 0 ms] (14) typed CpxTrs (15) RewriteLemmaProof [LOWER BOUND(ID), 378 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), 43 ms] (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 59 ms] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 50 ms] (26) typed CpxTrs (27) RewriteLemmaProof [LOWER BOUND(ID), 93 ms] (28) typed CpxTrs (29) RewriteLemmaProof [LOWER BOUND(ID), 58 ms] (30) typed CpxTrs (31) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (32) CpxWeightedTrs (33) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CpxWeightedTrs (35) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CpxTypedWeightedTrs (37) CompletionProof [UPPER BOUND(ID), 0 ms] (38) CpxTypedWeightedCompleteTrs (39) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CpxTypedWeightedCompleteTrs (41) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CpxRNTS (45) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CpxRNTS (47) ResultPropagationProof [UPPER BOUND(ID), 1 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 390 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 140 ms] (52) CpxRNTS (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 255 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 116 ms] (58) CpxRNTS (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 314 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 66 ms] (64) CpxRNTS (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (66) CpxRNTS (67) IntTrsBoundProof [UPPER BOUND(ID), 9595 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 2196 ms] (70) CpxRNTS (71) CompletionProof [UPPER BOUND(ID), 0 ms] (72) CpxTypedWeightedCompleteTrs (73) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (74) CpxRNTS (75) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (76) CdtProblem (77) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CpxRelTRS (83) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (84) CpxTRS (85) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CpxWeightedTrs (87) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CpxWeightedTrs (89) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CpxTypedWeightedTrs (91) CompletionProof [UPPER BOUND(ID), 0 ms] (92) CpxTypedWeightedCompleteTrs (93) NarrowingProof [BOTH BOUNDS(ID, ID), 513 ms] (94) CpxTypedWeightedCompleteTrs (95) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (96) CpxRNTS (97) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CpxRNTS (99) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CpxRNTS (101) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (102) CpxRNTS (103) IntTrsBoundProof [UPPER BOUND(ID), 201 ms] (104) CpxRNTS (105) IntTrsBoundProof [UPPER BOUND(ID), 54 ms] (106) CpxRNTS (107) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (108) CpxRNTS (109) IntTrsBoundProof [UPPER BOUND(ID), 553 ms] (110) CpxRNTS (111) IntTrsBoundProof [UPPER BOUND(ID), 66 ms] (112) CpxRNTS (113) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (114) CpxRNTS (115) IntTrsBoundProof [UPPER BOUND(ID), 170 ms] (116) CpxRNTS (117) IntTrsBoundProof [UPPER BOUND(ID), 85 ms] (118) CpxRNTS (119) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (120) CpxRNTS (121) IntTrsBoundProof [UPPER BOUND(ID), 372 ms] (122) CpxRNTS (123) IntTrsBoundProof [UPPER BOUND(ID), 86 ms] (124) CpxRNTS (125) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (126) CpxRNTS (127) IntTrsBoundProof [UPPER BOUND(ID), 333 ms] (128) CpxRNTS (129) IntTrsBoundProof [UPPER BOUND(ID), 106 ms] (130) CpxRNTS (131) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (132) CpxRNTS (133) IntTrsBoundProof [UPPER BOUND(ID), 140 ms] (134) CpxRNTS (135) IntTrsBoundProof [UPPER BOUND(ID), 104 ms] (136) CpxRNTS (137) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (138) CpxRNTS (139) IntTrsBoundProof [UPPER BOUND(ID), 3985 ms] (140) CpxRNTS (141) IntTrsBoundProof [UPPER BOUND(ID), 2306 ms] (142) CpxRNTS (143) CompletionProof [UPPER BOUND(ID), 0 ms] (144) CpxTypedWeightedCompleteTrs (145) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (146) CpxRNTS (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms] (148) CdtProblem (149) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 3 ms] (150) CdtProblem (151) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 10 ms] (190) CdtProblem (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 6 ms] (198) CdtProblem (199) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 25 ms] (200) CdtProblem (201) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms] (206) CdtProblem (207) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms] (208) CdtProblem (209) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 13 ms] (210) CdtProblem (211) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (212) CdtProblem (213) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (216) CdtProblem (217) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (218) CdtProblem (219) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (220) CdtProblem (221) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (222) CdtProblem (223) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (224) CdtProblem (225) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 18 ms] (226) CdtProblem (227) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (228) CdtProblem (229) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 4 ms] (230) CdtProblem (231) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (232) CdtProblem (233) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (234) CdtProblem (235) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 15 ms] (236) CdtProblem (237) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 39 ms] (238) CdtProblem (239) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (240) CdtProblem (241) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 35 ms] (242) CdtProblem (243) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 26 ms] (244) CdtProblem (245) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 11 ms] (246) CdtProblem (247) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 31 ms] (248) CdtProblem (249) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (250) CdtProblem (251) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 47 ms] (252) CdtProblem (253) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (254) CdtProblem (255) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms] (256) CdtProblem (257) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 13 ms] (258) CdtProblem (259) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 32 ms] (260) CdtProblem (261) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (262) CdtProblem (263) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 47 ms] (264) CdtProblem (265) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (266) CdtProblem (267) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 3 ms] (268) CdtProblem ---------------------------------------- (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)) -(x, 0) -> x -(s(x), s(y)) -> -(x, y) gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z) gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z))) gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z))) gcd(x, 0, 0) -> x gcd(0, y, 0) -> y gcd(0, 0, z) -> z 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)) -(x, 0') -> x -(s(x), s(y)) -> -(x, y) gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z) gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z))) gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z))) gcd(x, 0', 0') -> x gcd(0', y, 0') -> y gcd(0', 0', z) -> z 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)) -(x, 0) -> x -(s(x), s(y)) -> -(x, y) gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z) gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z))) gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z))) gcd(x, 0, 0) -> x gcd(0, y, 0) -> y gcd(0, 0, z) -> z 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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2) gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))) gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))) gcd(z0, 0, 0) -> z0 gcd(0, z0, 0) -> z0 gcd(0, 0, 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)) -'(z0, 0) -> c6 -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(z0, 0, 0) -> c17 GCD(0, z0, 0) -> c18 GCD(0, 0, z0) -> c19 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)) -'(z0, 0) -> c6 -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(z0, 0, 0) -> c17 GCD(0, z0, 0) -> c18 GCD(0, 0, z0) -> c19 K tuples:none Defined Rule Symbols: min_2, max_2, -_2, gcd_3 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c, c1, c2_1, c3, c4, c5_1, c6, c7_1, c8_3, c9_3, c10_2, c11_3, c12_3, c13_2, c14_3, c15_3, c16_2, c17, c18, c19 ---------------------------------------- (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)) -'(z0, 0) -> c6 -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(z0, 0, 0) -> c17 GCD(0, z0, 0) -> c18 GCD(0, 0, z0) -> c19 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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2) gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))) gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))) gcd(z0, 0, 0) -> z0 gcd(0, z0, 0) -> z0 gcd(0, 0, 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)) -'(z0, 0') -> c6 -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(z0, 0', 0') -> c17 GCD(0', z0, 0') -> c18 GCD(0', 0', z0) -> c19 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)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2) gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))) gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))) gcd(z0, 0', 0') -> z0 gcd(0', z0, 0') -> z0 gcd(0', 0', 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)) -'(z0, 0') -> c6 -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(z0, 0', 0') -> c17 GCD(0', z0, 0') -> c18 GCD(0', 0', z0) -> c19 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)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2) gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))) gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))) gcd(z0, 0', 0') -> z0 gcd(0', z0, 0') -> z0 gcd(0', 0', 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 -' :: 0':s -> 0':s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c6:c7 GCD :: 0':s -> 0':s -> 0':s -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c8 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 - :: 0':s -> 0':s -> 0':s max :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s c9 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c10 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c11 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c12 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c13 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c14 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c15 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c16 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c17 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c18 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c19 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 gcd :: 0':s -> 0':s -> 0':s -> 0':s hole_c:c1:c21_20 :: c:c1:c2 hole_0':s2_20 :: 0':s hole_c3:c4:c53_20 :: c3:c4:c5 hole_c6:c74_20 :: c6:c7 hole_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c195_20 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 gen_c:c1:c26_20 :: Nat -> c:c1:c2 gen_0':s7_20 :: Nat -> 0':s gen_c3:c4:c58_20 :: Nat -> c3:c4:c5 gen_c6:c79_20 :: Nat -> c6:c7 gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20 :: Nat -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 ---------------------------------------- (13) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: MIN, MAX, -', GCD, -, max, min, gcd They will be analysed ascendingly in the following order: MIN < GCD MAX < GCD -' < GCD - < GCD max < GCD min < GCD - < 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)) -'(z0, 0') -> c6 -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(z0, 0', 0') -> c17 GCD(0', z0, 0') -> c18 GCD(0', 0', z0) -> c19 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)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2) gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))) gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))) gcd(z0, 0', 0') -> z0 gcd(0', z0, 0') -> z0 gcd(0', 0', 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 -' :: 0':s -> 0':s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c6:c7 GCD :: 0':s -> 0':s -> 0':s -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c8 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 - :: 0':s -> 0':s -> 0':s max :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s c9 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c10 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c11 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c12 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c13 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c14 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c15 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c16 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c17 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c18 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c19 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 gcd :: 0':s -> 0':s -> 0':s -> 0':s hole_c:c1:c21_20 :: c:c1:c2 hole_0':s2_20 :: 0':s hole_c3:c4:c53_20 :: c3:c4:c5 hole_c6:c74_20 :: c6:c7 hole_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c195_20 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 gen_c:c1:c26_20 :: Nat -> c:c1:c2 gen_0':s7_20 :: Nat -> 0':s gen_c3:c4:c58_20 :: Nat -> c3:c4:c5 gen_c6:c79_20 :: Nat -> c6:c7 gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20 :: Nat -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 Generator Equations: gen_c:c1:c26_20(0) <=> c gen_c:c1:c26_20(+(x, 1)) <=> c2(gen_c:c1:c26_20(x)) gen_0':s7_20(0) <=> 0' gen_0':s7_20(+(x, 1)) <=> s(gen_0':s7_20(x)) gen_c3:c4:c58_20(0) <=> c3 gen_c3:c4:c58_20(+(x, 1)) <=> c5(gen_c3:c4:c58_20(x)) gen_c6:c79_20(0) <=> c6 gen_c6:c79_20(+(x, 1)) <=> c7(gen_c6:c79_20(x)) gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(0) <=> c17 gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(+(x, 1)) <=> c8(gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(x), c6, c3) The following defined symbols remain to be analysed: MIN, MAX, -', GCD, -, max, min, gcd They will be analysed ascendingly in the following order: MIN < GCD MAX < GCD -' < GCD - < GCD max < GCD min < GCD - < gcd max < gcd min < gcd ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: MIN(gen_0':s7_20(n12_20), gen_0':s7_20(n12_20)) -> gen_c:c1:c26_20(n12_20), rt in Omega(1 + n12_20) Induction Base: MIN(gen_0':s7_20(0), gen_0':s7_20(0)) ->_R^Omega(1) c Induction Step: MIN(gen_0':s7_20(+(n12_20, 1)), gen_0':s7_20(+(n12_20, 1))) ->_R^Omega(1) c2(MIN(gen_0':s7_20(n12_20), gen_0':s7_20(n12_20))) ->_IH c2(gen_c:c1:c26_20(c13_20)) 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)) -'(z0, 0') -> c6 -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(z0, 0', 0') -> c17 GCD(0', z0, 0') -> c18 GCD(0', 0', z0) -> c19 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)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2) gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))) gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))) gcd(z0, 0', 0') -> z0 gcd(0', z0, 0') -> z0 gcd(0', 0', 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 -' :: 0':s -> 0':s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c6:c7 GCD :: 0':s -> 0':s -> 0':s -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c8 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 - :: 0':s -> 0':s -> 0':s max :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s c9 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c10 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c11 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c12 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c13 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c14 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c15 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c16 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c17 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c18 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c19 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 gcd :: 0':s -> 0':s -> 0':s -> 0':s hole_c:c1:c21_20 :: c:c1:c2 hole_0':s2_20 :: 0':s hole_c3:c4:c53_20 :: c3:c4:c5 hole_c6:c74_20 :: c6:c7 hole_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c195_20 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 gen_c:c1:c26_20 :: Nat -> c:c1:c2 gen_0':s7_20 :: Nat -> 0':s gen_c3:c4:c58_20 :: Nat -> c3:c4:c5 gen_c6:c79_20 :: Nat -> c6:c7 gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20 :: Nat -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 Generator Equations: gen_c:c1:c26_20(0) <=> c gen_c:c1:c26_20(+(x, 1)) <=> c2(gen_c:c1:c26_20(x)) gen_0':s7_20(0) <=> 0' gen_0':s7_20(+(x, 1)) <=> s(gen_0':s7_20(x)) gen_c3:c4:c58_20(0) <=> c3 gen_c3:c4:c58_20(+(x, 1)) <=> c5(gen_c3:c4:c58_20(x)) gen_c6:c79_20(0) <=> c6 gen_c6:c79_20(+(x, 1)) <=> c7(gen_c6:c79_20(x)) gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(0) <=> c17 gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(+(x, 1)) <=> c8(gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(x), c6, c3) The following defined symbols remain to be analysed: MIN, MAX, -', GCD, -, max, min, gcd They will be analysed ascendingly in the following order: MIN < GCD MAX < GCD -' < GCD - < GCD max < GCD min < GCD - < 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)) -'(z0, 0') -> c6 -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(z0, 0', 0') -> c17 GCD(0', z0, 0') -> c18 GCD(0', 0', z0) -> c19 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)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2) gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))) gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))) gcd(z0, 0', 0') -> z0 gcd(0', z0, 0') -> z0 gcd(0', 0', 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 -' :: 0':s -> 0':s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c6:c7 GCD :: 0':s -> 0':s -> 0':s -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c8 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 - :: 0':s -> 0':s -> 0':s max :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s c9 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c10 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c11 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c12 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c13 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c14 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c15 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c16 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c17 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c18 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c19 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 gcd :: 0':s -> 0':s -> 0':s -> 0':s hole_c:c1:c21_20 :: c:c1:c2 hole_0':s2_20 :: 0':s hole_c3:c4:c53_20 :: c3:c4:c5 hole_c6:c74_20 :: c6:c7 hole_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c195_20 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 gen_c:c1:c26_20 :: Nat -> c:c1:c2 gen_0':s7_20 :: Nat -> 0':s gen_c3:c4:c58_20 :: Nat -> c3:c4:c5 gen_c6:c79_20 :: Nat -> c6:c7 gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20 :: Nat -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 Lemmas: MIN(gen_0':s7_20(n12_20), gen_0':s7_20(n12_20)) -> gen_c:c1:c26_20(n12_20), rt in Omega(1 + n12_20) Generator Equations: gen_c:c1:c26_20(0) <=> c gen_c:c1:c26_20(+(x, 1)) <=> c2(gen_c:c1:c26_20(x)) gen_0':s7_20(0) <=> 0' gen_0':s7_20(+(x, 1)) <=> s(gen_0':s7_20(x)) gen_c3:c4:c58_20(0) <=> c3 gen_c3:c4:c58_20(+(x, 1)) <=> c5(gen_c3:c4:c58_20(x)) gen_c6:c79_20(0) <=> c6 gen_c6:c79_20(+(x, 1)) <=> c7(gen_c6:c79_20(x)) gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(0) <=> c17 gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(+(x, 1)) <=> c8(gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(x), c6, c3) The following defined symbols remain to be analysed: MAX, -', GCD, -, max, min, gcd They will be analysed ascendingly in the following order: MAX < GCD -' < GCD - < GCD max < GCD min < GCD - < gcd max < gcd min < gcd ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: MAX(gen_0':s7_20(n596_20), gen_0':s7_20(n596_20)) -> gen_c3:c4:c58_20(n596_20), rt in Omega(1 + n596_20) Induction Base: MAX(gen_0':s7_20(0), gen_0':s7_20(0)) ->_R^Omega(1) c3 Induction Step: MAX(gen_0':s7_20(+(n596_20, 1)), gen_0':s7_20(+(n596_20, 1))) ->_R^Omega(1) c5(MAX(gen_0':s7_20(n596_20), gen_0':s7_20(n596_20))) ->_IH c5(gen_c3:c4:c58_20(c597_20)) 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)) -'(z0, 0') -> c6 -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(z0, 0', 0') -> c17 GCD(0', z0, 0') -> c18 GCD(0', 0', z0) -> c19 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)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2) gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))) gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))) gcd(z0, 0', 0') -> z0 gcd(0', z0, 0') -> z0 gcd(0', 0', 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 -' :: 0':s -> 0':s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c6:c7 GCD :: 0':s -> 0':s -> 0':s -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c8 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 - :: 0':s -> 0':s -> 0':s max :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s c9 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c10 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c11 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c12 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c13 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c14 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c15 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c16 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c17 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c18 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c19 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 gcd :: 0':s -> 0':s -> 0':s -> 0':s hole_c:c1:c21_20 :: c:c1:c2 hole_0':s2_20 :: 0':s hole_c3:c4:c53_20 :: c3:c4:c5 hole_c6:c74_20 :: c6:c7 hole_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c195_20 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 gen_c:c1:c26_20 :: Nat -> c:c1:c2 gen_0':s7_20 :: Nat -> 0':s gen_c3:c4:c58_20 :: Nat -> c3:c4:c5 gen_c6:c79_20 :: Nat -> c6:c7 gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20 :: Nat -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 Lemmas: MIN(gen_0':s7_20(n12_20), gen_0':s7_20(n12_20)) -> gen_c:c1:c26_20(n12_20), rt in Omega(1 + n12_20) MAX(gen_0':s7_20(n596_20), gen_0':s7_20(n596_20)) -> gen_c3:c4:c58_20(n596_20), rt in Omega(1 + n596_20) Generator Equations: gen_c:c1:c26_20(0) <=> c gen_c:c1:c26_20(+(x, 1)) <=> c2(gen_c:c1:c26_20(x)) gen_0':s7_20(0) <=> 0' gen_0':s7_20(+(x, 1)) <=> s(gen_0':s7_20(x)) gen_c3:c4:c58_20(0) <=> c3 gen_c3:c4:c58_20(+(x, 1)) <=> c5(gen_c3:c4:c58_20(x)) gen_c6:c79_20(0) <=> c6 gen_c6:c79_20(+(x, 1)) <=> c7(gen_c6:c79_20(x)) gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(0) <=> c17 gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(+(x, 1)) <=> c8(gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(x), c6, c3) The following defined symbols remain to be analysed: -', GCD, -, max, min, gcd They will be analysed ascendingly in the following order: -' < GCD - < GCD max < GCD min < GCD - < gcd max < gcd min < gcd ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: -'(gen_0':s7_20(n1255_20), gen_0':s7_20(n1255_20)) -> gen_c6:c79_20(n1255_20), rt in Omega(1 + n1255_20) Induction Base: -'(gen_0':s7_20(0), gen_0':s7_20(0)) ->_R^Omega(1) c6 Induction Step: -'(gen_0':s7_20(+(n1255_20, 1)), gen_0':s7_20(+(n1255_20, 1))) ->_R^Omega(1) c7(-'(gen_0':s7_20(n1255_20), gen_0':s7_20(n1255_20))) ->_IH c7(gen_c6:c79_20(c1256_20)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (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)) -'(z0, 0') -> c6 -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(z0, 0', 0') -> c17 GCD(0', z0, 0') -> c18 GCD(0', 0', z0) -> c19 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)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2) gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))) gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))) gcd(z0, 0', 0') -> z0 gcd(0', z0, 0') -> z0 gcd(0', 0', 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 -' :: 0':s -> 0':s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c6:c7 GCD :: 0':s -> 0':s -> 0':s -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c8 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 - :: 0':s -> 0':s -> 0':s max :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s c9 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c10 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c11 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c12 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c13 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c14 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c15 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c16 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c17 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c18 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c19 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 gcd :: 0':s -> 0':s -> 0':s -> 0':s hole_c:c1:c21_20 :: c:c1:c2 hole_0':s2_20 :: 0':s hole_c3:c4:c53_20 :: c3:c4:c5 hole_c6:c74_20 :: c6:c7 hole_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c195_20 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 gen_c:c1:c26_20 :: Nat -> c:c1:c2 gen_0':s7_20 :: Nat -> 0':s gen_c3:c4:c58_20 :: Nat -> c3:c4:c5 gen_c6:c79_20 :: Nat -> c6:c7 gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20 :: Nat -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 Lemmas: MIN(gen_0':s7_20(n12_20), gen_0':s7_20(n12_20)) -> gen_c:c1:c26_20(n12_20), rt in Omega(1 + n12_20) MAX(gen_0':s7_20(n596_20), gen_0':s7_20(n596_20)) -> gen_c3:c4:c58_20(n596_20), rt in Omega(1 + n596_20) -'(gen_0':s7_20(n1255_20), gen_0':s7_20(n1255_20)) -> gen_c6:c79_20(n1255_20), rt in Omega(1 + n1255_20) Generator Equations: gen_c:c1:c26_20(0) <=> c gen_c:c1:c26_20(+(x, 1)) <=> c2(gen_c:c1:c26_20(x)) gen_0':s7_20(0) <=> 0' gen_0':s7_20(+(x, 1)) <=> s(gen_0':s7_20(x)) gen_c3:c4:c58_20(0) <=> c3 gen_c3:c4:c58_20(+(x, 1)) <=> c5(gen_c3:c4:c58_20(x)) gen_c6:c79_20(0) <=> c6 gen_c6:c79_20(+(x, 1)) <=> c7(gen_c6:c79_20(x)) gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(0) <=> c17 gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(+(x, 1)) <=> c8(gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(x), c6, c3) The following defined symbols remain to be analysed: -, GCD, max, min, gcd They will be analysed ascendingly in the following order: - < GCD max < GCD min < GCD - < gcd max < gcd min < gcd ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: -(gen_0':s7_20(n1924_20), gen_0':s7_20(n1924_20)) -> gen_0':s7_20(0), rt in Omega(0) Induction Base: -(gen_0':s7_20(0), gen_0':s7_20(0)) ->_R^Omega(0) gen_0':s7_20(0) Induction Step: -(gen_0':s7_20(+(n1924_20, 1)), gen_0':s7_20(+(n1924_20, 1))) ->_R^Omega(0) -(gen_0':s7_20(n1924_20), gen_0':s7_20(n1924_20)) ->_IH gen_0':s7_20(0) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (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)) -'(z0, 0') -> c6 -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(z0, 0', 0') -> c17 GCD(0', z0, 0') -> c18 GCD(0', 0', z0) -> c19 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)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2) gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))) gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))) gcd(z0, 0', 0') -> z0 gcd(0', z0, 0') -> z0 gcd(0', 0', 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 -' :: 0':s -> 0':s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c6:c7 GCD :: 0':s -> 0':s -> 0':s -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c8 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 - :: 0':s -> 0':s -> 0':s max :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s c9 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c10 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c11 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c12 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c13 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c14 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c15 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c16 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c17 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c18 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c19 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 gcd :: 0':s -> 0':s -> 0':s -> 0':s hole_c:c1:c21_20 :: c:c1:c2 hole_0':s2_20 :: 0':s hole_c3:c4:c53_20 :: c3:c4:c5 hole_c6:c74_20 :: c6:c7 hole_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c195_20 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 gen_c:c1:c26_20 :: Nat -> c:c1:c2 gen_0':s7_20 :: Nat -> 0':s gen_c3:c4:c58_20 :: Nat -> c3:c4:c5 gen_c6:c79_20 :: Nat -> c6:c7 gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20 :: Nat -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 Lemmas: MIN(gen_0':s7_20(n12_20), gen_0':s7_20(n12_20)) -> gen_c:c1:c26_20(n12_20), rt in Omega(1 + n12_20) MAX(gen_0':s7_20(n596_20), gen_0':s7_20(n596_20)) -> gen_c3:c4:c58_20(n596_20), rt in Omega(1 + n596_20) -'(gen_0':s7_20(n1255_20), gen_0':s7_20(n1255_20)) -> gen_c6:c79_20(n1255_20), rt in Omega(1 + n1255_20) -(gen_0':s7_20(n1924_20), gen_0':s7_20(n1924_20)) -> gen_0':s7_20(0), rt in Omega(0) Generator Equations: gen_c:c1:c26_20(0) <=> c gen_c:c1:c26_20(+(x, 1)) <=> c2(gen_c:c1:c26_20(x)) gen_0':s7_20(0) <=> 0' gen_0':s7_20(+(x, 1)) <=> s(gen_0':s7_20(x)) gen_c3:c4:c58_20(0) <=> c3 gen_c3:c4:c58_20(+(x, 1)) <=> c5(gen_c3:c4:c58_20(x)) gen_c6:c79_20(0) <=> c6 gen_c6:c79_20(+(x, 1)) <=> c7(gen_c6:c79_20(x)) gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(0) <=> c17 gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(+(x, 1)) <=> c8(gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(x), c6, c3) The following defined symbols remain to be analysed: max, GCD, min, gcd They will be analysed ascendingly in the following order: max < GCD min < GCD max < gcd min < gcd ---------------------------------------- (27) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: max(gen_0':s7_20(n2800_20), gen_0':s7_20(n2800_20)) -> gen_0':s7_20(n2800_20), rt in Omega(0) Induction Base: max(gen_0':s7_20(0), gen_0':s7_20(0)) ->_R^Omega(0) gen_0':s7_20(0) Induction Step: max(gen_0':s7_20(+(n2800_20, 1)), gen_0':s7_20(+(n2800_20, 1))) ->_R^Omega(0) s(max(gen_0':s7_20(n2800_20), gen_0':s7_20(n2800_20))) ->_IH s(gen_0':s7_20(c2801_20)) 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)) -'(z0, 0') -> c6 -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(z0, 0', 0') -> c17 GCD(0', z0, 0') -> c18 GCD(0', 0', z0) -> c19 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)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2) gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))) gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))) gcd(z0, 0', 0') -> z0 gcd(0', z0, 0') -> z0 gcd(0', 0', 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 -' :: 0':s -> 0':s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c6:c7 GCD :: 0':s -> 0':s -> 0':s -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c8 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 - :: 0':s -> 0':s -> 0':s max :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s c9 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c10 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c11 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c12 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c13 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c14 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c15 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c16 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c17 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c18 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c19 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 gcd :: 0':s -> 0':s -> 0':s -> 0':s hole_c:c1:c21_20 :: c:c1:c2 hole_0':s2_20 :: 0':s hole_c3:c4:c53_20 :: c3:c4:c5 hole_c6:c74_20 :: c6:c7 hole_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c195_20 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 gen_c:c1:c26_20 :: Nat -> c:c1:c2 gen_0':s7_20 :: Nat -> 0':s gen_c3:c4:c58_20 :: Nat -> c3:c4:c5 gen_c6:c79_20 :: Nat -> c6:c7 gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20 :: Nat -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 Lemmas: MIN(gen_0':s7_20(n12_20), gen_0':s7_20(n12_20)) -> gen_c:c1:c26_20(n12_20), rt in Omega(1 + n12_20) MAX(gen_0':s7_20(n596_20), gen_0':s7_20(n596_20)) -> gen_c3:c4:c58_20(n596_20), rt in Omega(1 + n596_20) -'(gen_0':s7_20(n1255_20), gen_0':s7_20(n1255_20)) -> gen_c6:c79_20(n1255_20), rt in Omega(1 + n1255_20) -(gen_0':s7_20(n1924_20), gen_0':s7_20(n1924_20)) -> gen_0':s7_20(0), rt in Omega(0) max(gen_0':s7_20(n2800_20), gen_0':s7_20(n2800_20)) -> gen_0':s7_20(n2800_20), rt in Omega(0) Generator Equations: gen_c:c1:c26_20(0) <=> c gen_c:c1:c26_20(+(x, 1)) <=> c2(gen_c:c1:c26_20(x)) gen_0':s7_20(0) <=> 0' gen_0':s7_20(+(x, 1)) <=> s(gen_0':s7_20(x)) gen_c3:c4:c58_20(0) <=> c3 gen_c3:c4:c58_20(+(x, 1)) <=> c5(gen_c3:c4:c58_20(x)) gen_c6:c79_20(0) <=> c6 gen_c6:c79_20(+(x, 1)) <=> c7(gen_c6:c79_20(x)) gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(0) <=> c17 gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(+(x, 1)) <=> c8(gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(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':s7_20(n3770_20), gen_0':s7_20(n3770_20)) -> gen_0':s7_20(n3770_20), rt in Omega(0) Induction Base: min(gen_0':s7_20(0), gen_0':s7_20(0)) ->_R^Omega(0) 0' Induction Step: min(gen_0':s7_20(+(n3770_20, 1)), gen_0':s7_20(+(n3770_20, 1))) ->_R^Omega(0) s(min(gen_0':s7_20(n3770_20), gen_0':s7_20(n3770_20))) ->_IH s(gen_0':s7_20(c3771_20)) 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)) -'(z0, 0') -> c6 -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(z0, 0', 0') -> c17 GCD(0', z0, 0') -> c18 GCD(0', 0', z0) -> c19 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)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2) gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))) gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))) gcd(z0, 0', 0') -> z0 gcd(0', z0, 0') -> z0 gcd(0', 0', 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 -' :: 0':s -> 0':s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c6:c7 GCD :: 0':s -> 0':s -> 0':s -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c8 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 - :: 0':s -> 0':s -> 0':s max :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s c9 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c10 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c11 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c12 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c13 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c14 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c3:c4:c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c15 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c6:c7 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c16 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c:c1:c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c17 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c18 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 c19 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 gcd :: 0':s -> 0':s -> 0':s -> 0':s hole_c:c1:c21_20 :: c:c1:c2 hole_0':s2_20 :: 0':s hole_c3:c4:c53_20 :: c3:c4:c5 hole_c6:c74_20 :: c6:c7 hole_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c195_20 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 gen_c:c1:c26_20 :: Nat -> c:c1:c2 gen_0':s7_20 :: Nat -> 0':s gen_c3:c4:c58_20 :: Nat -> c3:c4:c5 gen_c6:c79_20 :: Nat -> c6:c7 gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20 :: Nat -> c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 Lemmas: MIN(gen_0':s7_20(n12_20), gen_0':s7_20(n12_20)) -> gen_c:c1:c26_20(n12_20), rt in Omega(1 + n12_20) MAX(gen_0':s7_20(n596_20), gen_0':s7_20(n596_20)) -> gen_c3:c4:c58_20(n596_20), rt in Omega(1 + n596_20) -'(gen_0':s7_20(n1255_20), gen_0':s7_20(n1255_20)) -> gen_c6:c79_20(n1255_20), rt in Omega(1 + n1255_20) -(gen_0':s7_20(n1924_20), gen_0':s7_20(n1924_20)) -> gen_0':s7_20(0), rt in Omega(0) max(gen_0':s7_20(n2800_20), gen_0':s7_20(n2800_20)) -> gen_0':s7_20(n2800_20), rt in Omega(0) min(gen_0':s7_20(n3770_20), gen_0':s7_20(n3770_20)) -> gen_0':s7_20(n3770_20), rt in Omega(0) Generator Equations: gen_c:c1:c26_20(0) <=> c gen_c:c1:c26_20(+(x, 1)) <=> c2(gen_c:c1:c26_20(x)) gen_0':s7_20(0) <=> 0' gen_0':s7_20(+(x, 1)) <=> s(gen_0':s7_20(x)) gen_c3:c4:c58_20(0) <=> c3 gen_c3:c4:c58_20(+(x, 1)) <=> c5(gen_c3:c4:c58_20(x)) gen_c6:c79_20(0) <=> c6 gen_c6:c79_20(+(x, 1)) <=> c7(gen_c6:c79_20(x)) gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(0) <=> c17 gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(+(x, 1)) <=> c8(gen_c8:c9:c10:c11:c12:c13:c14:c15:c16:c17:c18:c1910_20(x), c6, c3) The following defined symbols remain to be analysed: GCD, gcd ---------------------------------------- (31) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (32) 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] -(x, 0) -> x [1] -(s(x), s(y)) -> -(x, y) [1] gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z) [1] gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z))) [1] gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z))) [1] gcd(x, 0, 0) -> x [1] gcd(0, y, 0) -> y [1] gcd(0, 0, z) -> z [1] Rewrite Strategy: INNERMOST ---------------------------------------- (33) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: - => minus ---------------------------------------- (34) 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)) -> minus(x, y) [1] gcd(s(x), s(y), z) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)), z) [1] gcd(x, s(y), s(z)) -> gcd(x, minus(max(y, z), min(y, z)), s(min(y, z))) [1] gcd(s(x), y, s(z)) -> gcd(minus(max(x, z), min(x, z)), y, s(min(x, z))) [1] gcd(x, 0, 0) -> x [1] gcd(0, y, 0) -> y [1] gcd(0, 0, z) -> z [1] Rewrite Strategy: INNERMOST ---------------------------------------- (35) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (36) 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)) -> minus(x, y) [1] gcd(s(x), s(y), z) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)), z) [1] gcd(x, s(y), s(z)) -> gcd(x, minus(max(y, z), min(y, z)), s(min(y, z))) [1] gcd(s(x), y, s(z)) -> gcd(minus(max(x, z), min(x, z)), y, s(min(x, z))) [1] gcd(x, 0, 0) -> x [1] gcd(0, y, 0) -> y [1] gcd(0, 0, z) -> z [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 gcd :: 0:s -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (37) 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_3 (c) The following functions are completely defined: minus_2 max_2 min_2 Due to the following rules being added: minus(v0, v1) -> 0 [0] And the following fresh constants: none ---------------------------------------- (38) 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)) -> minus(x, y) [1] gcd(s(x), s(y), z) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)), z) [1] gcd(x, s(y), s(z)) -> gcd(x, minus(max(y, z), min(y, z)), s(min(y, z))) [1] gcd(s(x), y, s(z)) -> gcd(minus(max(x, z), min(x, z)), y, s(min(x, z))) [1] gcd(x, 0, 0) -> x [1] gcd(0, y, 0) -> y [1] gcd(0, 0, z) -> z [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 gcd :: 0:s -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (39) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (40) 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)) -> minus(x, y) [1] gcd(s(x), s(0), z) -> gcd(minus(x, 0), s(0), z) [4] gcd(s(0), s(0), z) -> gcd(minus(0, 0), s(0), z) [4] gcd(s(0), s(0), z) -> gcd(minus(0, 0), s(0), z) [4] gcd(s(0), s(0), z) -> gcd(minus(0, 0), s(0), z) [4] gcd(s(0), s(0), z) -> gcd(minus(0, 0), s(0), z) [4] gcd(s(0), s(0), z) -> gcd(minus(0, 0), s(0), z) [4] gcd(s(0), s(0), z) -> gcd(minus(0, 0), s(0), z) [4] gcd(s(0), s(y), z) -> gcd(minus(y, 0), s(0), z) [4] gcd(s(s(x')), s(s(y')), z) -> gcd(minus(s(max(x', y')), s(min(x', y'))), s(s(min(x', y'))), z) [4] gcd(x, s(y), s(0)) -> gcd(x, minus(y, 0), s(0)) [4] gcd(x, s(0), s(0)) -> gcd(x, minus(0, 0), s(0)) [4] gcd(x, s(0), s(0)) -> gcd(x, minus(0, 0), s(0)) [4] gcd(x, s(0), s(0)) -> gcd(x, minus(0, 0), s(0)) [4] gcd(x, s(0), s(0)) -> gcd(x, minus(0, 0), s(0)) [4] gcd(x, s(0), s(0)) -> gcd(x, minus(0, 0), s(0)) [4] gcd(x, s(0), s(0)) -> gcd(x, minus(0, 0), s(0)) [4] gcd(x, s(0), s(z)) -> gcd(x, minus(z, 0), s(0)) [4] gcd(x, s(s(x'')), s(s(y''))) -> gcd(x, minus(s(max(x'', y'')), s(min(x'', y''))), s(s(min(x'', y'')))) [4] gcd(s(x), y, s(0)) -> gcd(minus(x, 0), y, s(0)) [4] gcd(s(0), y, s(0)) -> gcd(minus(0, 0), y, s(0)) [4] gcd(s(0), y, s(0)) -> gcd(minus(0, 0), y, s(0)) [4] gcd(s(0), y, s(0)) -> gcd(minus(0, 0), y, s(0)) [4] gcd(s(0), y, s(0)) -> gcd(minus(0, 0), y, s(0)) [4] gcd(s(0), y, s(0)) -> gcd(minus(0, 0), y, s(0)) [4] gcd(s(0), y, s(0)) -> gcd(minus(0, 0), y, s(0)) [4] gcd(s(0), y, s(z)) -> gcd(minus(z, 0), y, s(0)) [4] gcd(s(s(x1)), y, s(s(y1))) -> gcd(minus(s(max(x1, y1)), s(min(x1, y1))), y, s(s(min(x1, y1)))) [4] gcd(x, 0, 0) -> x [1] gcd(0, y, 0) -> y [1] gcd(0, 0, z) -> z [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 gcd :: 0:s -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (41) 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 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: gcd(z', z'', z1) -{ 1 }-> x :|: z'' = 0, z1 = 0, z' = x, x >= 0 gcd(z', z'', z1) -{ 1 }-> y :|: z1 = 0, z'' = y, y >= 0, z' = 0 gcd(z', z'', z1) -{ 1 }-> z :|: z'' = 0, z1 = z, z >= 0, z' = 0 gcd(z', z'', z1) -{ 4 }-> gcd(x, minus(y, 0), 1 + 0) :|: z' = x, z1 = 1 + 0, x >= 0, y >= 0, z'' = 1 + y gcd(z', z'', z1) -{ 4 }-> gcd(x, minus(z, 0), 1 + 0) :|: z >= 0, z' = x, x >= 0, z1 = 1 + z, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(x, minus(0, 0), 1 + 0) :|: z' = x, z1 = 1 + 0, x >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(x, minus(1 + max(x'', y''), 1 + min(x'', y'')), 1 + (1 + min(x'', y''))) :|: z' = x, x >= 0, z'' = 1 + (1 + x''), y'' >= 0, z1 = 1 + (1 + y''), x'' >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(x, 0), y, 1 + 0) :|: z' = 1 + x, z'' = y, z1 = 1 + 0, x >= 0, y >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(x, 0), 1 + 0, z) :|: z' = 1 + x, z1 = z, z >= 0, x >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(y, 0), 1 + 0, z) :|: z1 = z, z >= 0, y >= 0, z'' = 1 + y, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z, 0), y, 1 + 0) :|: z >= 0, z'' = y, y >= 0, z' = 1 + 0, z1 = 1 + z gcd(z', z'', z1) -{ 4 }-> gcd(minus(0, 0), y, 1 + 0) :|: z'' = y, z1 = 1 + 0, y >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(0, 0), 1 + 0, z) :|: z1 = z, z >= 0, z' = 1 + 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(1 + max(x', y'), 1 + min(x', y')), 1 + (1 + min(x', y')), z) :|: z' = 1 + (1 + x'), z1 = z, z >= 0, x' >= 0, z'' = 1 + (1 + y'), y' >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(1 + max(x1, y1), 1 + min(x1, y1)), y, 1 + (1 + min(x1, y1))) :|: y1 >= 0, x1 >= 0, z' = 1 + (1 + x1), z'' = y, y >= 0, z1 = 1 + (1 + y1) max(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0 max(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0 max(z', z'') -{ 1 }-> 1 + max(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y min(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = x, x >= 0 min(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0 min(z', z'') -{ 1 }-> 1 + min(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y minus(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0 minus(z', z'') -{ 1 }-> minus(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y minus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 ---------------------------------------- (43) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: gcd(z', z'', z1) -{ 1 }-> z' :|: z'' = 0, z1 = 0, z' >= 0 gcd(z', z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0, z' = 0 gcd(z', z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0, z' = 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(0, 0), 1 + 0) :|: z1 = 1 + 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(z'' - 1, 0), 1 + 0) :|: z1 = 1 + 0, z' >= 0, z'' - 1 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(z1 - 1, 0), 1 + 0) :|: z1 - 1 >= 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(1 + max(z'' - 2, z1 - 2), 1 + min(z'' - 2, z1 - 2)), 1 + (1 + min(z'' - 2, z1 - 2))) :|: z' >= 0, z1 - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(0, 0), z'', 1 + 0) :|: z1 = 1 + 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(0, 0), 1 + 0, z1) :|: z1 >= 0, z' = 1 + 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z' - 1, 0), z'', 1 + 0) :|: z1 = 1 + 0, z' - 1 >= 0, z'' >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0, z1) :|: z1 >= 0, z' - 1 >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z'' - 1, 0), 1 + 0, z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z1 - 1, 0), z'', 1 + 0) :|: z1 - 1 >= 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(1 + max(z' - 2, z'' - 2), 1 + min(z' - 2, z'' - 2)), 1 + (1 + min(z' - 2, z'' - 2)), z1) :|: z1 >= 0, z' - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(1 + max(z' - 2, z1 - 2), 1 + min(z' - 2, z1 - 2)), z'', 1 + (1 + min(z' - 2, z1 - 2))) :|: z1 - 2 >= 0, z' - 2 >= 0, z'' >= 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'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 ---------------------------------------- (45) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { max } { minus } { min } { gcd } ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: gcd(z', z'', z1) -{ 1 }-> z' :|: z'' = 0, z1 = 0, z' >= 0 gcd(z', z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0, z' = 0 gcd(z', z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0, z' = 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(0, 0), 1 + 0) :|: z1 = 1 + 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(z'' - 1, 0), 1 + 0) :|: z1 = 1 + 0, z' >= 0, z'' - 1 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(z1 - 1, 0), 1 + 0) :|: z1 - 1 >= 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(1 + max(z'' - 2, z1 - 2), 1 + min(z'' - 2, z1 - 2)), 1 + (1 + min(z'' - 2, z1 - 2))) :|: z' >= 0, z1 - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(0, 0), z'', 1 + 0) :|: z1 = 1 + 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(0, 0), 1 + 0, z1) :|: z1 >= 0, z' = 1 + 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z' - 1, 0), z'', 1 + 0) :|: z1 = 1 + 0, z' - 1 >= 0, z'' >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0, z1) :|: z1 >= 0, z' - 1 >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z'' - 1, 0), 1 + 0, z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z1 - 1, 0), z'', 1 + 0) :|: z1 - 1 >= 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(1 + max(z' - 2, z'' - 2), 1 + min(z' - 2, z'' - 2)), 1 + (1 + min(z' - 2, z'' - 2)), z1) :|: z1 >= 0, z' - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(1 + max(z' - 2, z1 - 2), 1 + min(z' - 2, z1 - 2)), z'', 1 + (1 + min(z' - 2, z1 - 2))) :|: z1 - 2 >= 0, z' - 2 >= 0, z'' >= 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'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {max}, {minus}, {min}, {gcd} ---------------------------------------- (47) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: gcd(z', z'', z1) -{ 1 }-> z' :|: z'' = 0, z1 = 0, z' >= 0 gcd(z', z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0, z' = 0 gcd(z', z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0, z' = 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(0, 0), 1 + 0) :|: z1 = 1 + 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(z'' - 1, 0), 1 + 0) :|: z1 = 1 + 0, z' >= 0, z'' - 1 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(z1 - 1, 0), 1 + 0) :|: z1 - 1 >= 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(1 + max(z'' - 2, z1 - 2), 1 + min(z'' - 2, z1 - 2)), 1 + (1 + min(z'' - 2, z1 - 2))) :|: z' >= 0, z1 - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(0, 0), z'', 1 + 0) :|: z1 = 1 + 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(0, 0), 1 + 0, z1) :|: z1 >= 0, z' = 1 + 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z' - 1, 0), z'', 1 + 0) :|: z1 = 1 + 0, z' - 1 >= 0, z'' >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0, z1) :|: z1 >= 0, z' - 1 >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z'' - 1, 0), 1 + 0, z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z1 - 1, 0), z'', 1 + 0) :|: z1 - 1 >= 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(1 + max(z' - 2, z'' - 2), 1 + min(z' - 2, z'' - 2)), 1 + (1 + min(z' - 2, z'' - 2)), z1) :|: z1 >= 0, z' - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(1 + max(z' - 2, z1 - 2), 1 + min(z' - 2, z1 - 2)), z'', 1 + (1 + min(z' - 2, z1 - 2))) :|: z1 - 2 >= 0, z' - 2 >= 0, z'' >= 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'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {max}, {minus}, {min}, {gcd} ---------------------------------------- (49) 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'' ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: gcd(z', z'', z1) -{ 1 }-> z' :|: z'' = 0, z1 = 0, z' >= 0 gcd(z', z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0, z' = 0 gcd(z', z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0, z' = 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(0, 0), 1 + 0) :|: z1 = 1 + 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(z'' - 1, 0), 1 + 0) :|: z1 = 1 + 0, z' >= 0, z'' - 1 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(z1 - 1, 0), 1 + 0) :|: z1 - 1 >= 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(1 + max(z'' - 2, z1 - 2), 1 + min(z'' - 2, z1 - 2)), 1 + (1 + min(z'' - 2, z1 - 2))) :|: z' >= 0, z1 - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(0, 0), z'', 1 + 0) :|: z1 = 1 + 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(0, 0), 1 + 0, z1) :|: z1 >= 0, z' = 1 + 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z' - 1, 0), z'', 1 + 0) :|: z1 = 1 + 0, z' - 1 >= 0, z'' >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0, z1) :|: z1 >= 0, z' - 1 >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z'' - 1, 0), 1 + 0, z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z1 - 1, 0), z'', 1 + 0) :|: z1 - 1 >= 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(1 + max(z' - 2, z'' - 2), 1 + min(z' - 2, z'' - 2)), 1 + (1 + min(z' - 2, z'' - 2)), z1) :|: z1 >= 0, z' - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(1 + max(z' - 2, z1 - 2), 1 + min(z' - 2, z1 - 2)), z'', 1 + (1 + min(z' - 2, z1 - 2))) :|: z1 - 2 >= 0, z' - 2 >= 0, z'' >= 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'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {max}, {minus}, {min}, {gcd} Previous analysis results are: max: runtime: ?, size: O(n^1) [z' + z''] ---------------------------------------- (51) 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'' ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: gcd(z', z'', z1) -{ 1 }-> z' :|: z'' = 0, z1 = 0, z' >= 0 gcd(z', z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0, z' = 0 gcd(z', z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0, z' = 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(0, 0), 1 + 0) :|: z1 = 1 + 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(z'' - 1, 0), 1 + 0) :|: z1 = 1 + 0, z' >= 0, z'' - 1 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(z1 - 1, 0), 1 + 0) :|: z1 - 1 >= 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(1 + max(z'' - 2, z1 - 2), 1 + min(z'' - 2, z1 - 2)), 1 + (1 + min(z'' - 2, z1 - 2))) :|: z' >= 0, z1 - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(0, 0), z'', 1 + 0) :|: z1 = 1 + 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(0, 0), 1 + 0, z1) :|: z1 >= 0, z' = 1 + 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z' - 1, 0), z'', 1 + 0) :|: z1 = 1 + 0, z' - 1 >= 0, z'' >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0, z1) :|: z1 >= 0, z' - 1 >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z'' - 1, 0), 1 + 0, z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z1 - 1, 0), z'', 1 + 0) :|: z1 - 1 >= 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(1 + max(z' - 2, z'' - 2), 1 + min(z' - 2, z'' - 2)), 1 + (1 + min(z' - 2, z'' - 2)), z1) :|: z1 >= 0, z' - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(1 + max(z' - 2, z1 - 2), 1 + min(z' - 2, z1 - 2)), z'', 1 + (1 + min(z' - 2, z1 - 2))) :|: z1 - 2 >= 0, z' - 2 >= 0, z'' >= 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'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {minus}, {min}, {gcd} Previous analysis results are: max: runtime: O(n^1) [2 + z''], size: O(n^1) [z' + z''] ---------------------------------------- (53) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: gcd(z', z'', z1) -{ 1 }-> z' :|: z'' = 0, z1 = 0, z' >= 0 gcd(z', z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0, z' = 0 gcd(z', z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0, z' = 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(0, 0), 1 + 0) :|: z1 = 1 + 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(z'' - 1, 0), 1 + 0) :|: z1 = 1 + 0, z' >= 0, z'' - 1 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(z1 - 1, 0), 1 + 0) :|: z1 - 1 >= 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 + z1 }-> gcd(z', minus(1 + s'', 1 + min(z'' - 2, z1 - 2)), 1 + (1 + min(z'' - 2, z1 - 2))) :|: s'' >= 0, s'' <= z1 - 2 + (z'' - 2), z' >= 0, z1 - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(0, 0), z'', 1 + 0) :|: z1 = 1 + 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(0, 0), 1 + 0, z1) :|: z1 >= 0, z' = 1 + 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z' - 1, 0), z'', 1 + 0) :|: z1 = 1 + 0, z' - 1 >= 0, z'' >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0, z1) :|: z1 >= 0, z' - 1 >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z'' - 1, 0), 1 + 0, z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z1 - 1, 0), z'', 1 + 0) :|: z1 - 1 >= 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 + z'' }-> gcd(minus(1 + s', 1 + min(z' - 2, z'' - 2)), 1 + (1 + min(z' - 2, z'' - 2)), z1) :|: s' >= 0, s' <= z'' - 2 + (z' - 2), z1 >= 0, z' - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 + z1 }-> gcd(minus(1 + s1, 1 + min(z' - 2, z1 - 2)), z'', 1 + (1 + min(z' - 2, z1 - 2))) :|: s1 >= 0, s1 <= z1 - 2 + (z' - 2), z1 - 2 >= 0, z' - 2 >= 0, z'' >= 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'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {minus}, {min}, {gcd} Previous analysis results are: max: runtime: O(n^1) [2 + z''], size: O(n^1) [z' + z''] ---------------------------------------- (55) 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' ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: gcd(z', z'', z1) -{ 1 }-> z' :|: z'' = 0, z1 = 0, z' >= 0 gcd(z', z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0, z' = 0 gcd(z', z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0, z' = 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(0, 0), 1 + 0) :|: z1 = 1 + 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(z'' - 1, 0), 1 + 0) :|: z1 = 1 + 0, z' >= 0, z'' - 1 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(z1 - 1, 0), 1 + 0) :|: z1 - 1 >= 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 + z1 }-> gcd(z', minus(1 + s'', 1 + min(z'' - 2, z1 - 2)), 1 + (1 + min(z'' - 2, z1 - 2))) :|: s'' >= 0, s'' <= z1 - 2 + (z'' - 2), z' >= 0, z1 - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(0, 0), z'', 1 + 0) :|: z1 = 1 + 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(0, 0), 1 + 0, z1) :|: z1 >= 0, z' = 1 + 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z' - 1, 0), z'', 1 + 0) :|: z1 = 1 + 0, z' - 1 >= 0, z'' >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0, z1) :|: z1 >= 0, z' - 1 >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z'' - 1, 0), 1 + 0, z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z1 - 1, 0), z'', 1 + 0) :|: z1 - 1 >= 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 + z'' }-> gcd(minus(1 + s', 1 + min(z' - 2, z'' - 2)), 1 + (1 + min(z' - 2, z'' - 2)), z1) :|: s' >= 0, s' <= z'' - 2 + (z' - 2), z1 >= 0, z' - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 + z1 }-> gcd(minus(1 + s1, 1 + min(z' - 2, z1 - 2)), z'', 1 + (1 + min(z' - 2, z1 - 2))) :|: s1 >= 0, s1 <= z1 - 2 + (z' - 2), z1 - 2 >= 0, z' - 2 >= 0, z'' >= 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'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {minus}, {min}, {gcd} Previous analysis results are: max: runtime: O(n^1) [2 + z''], size: O(n^1) [z' + z''] minus: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z'' ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: gcd(z', z'', z1) -{ 1 }-> z' :|: z'' = 0, z1 = 0, z' >= 0 gcd(z', z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0, z' = 0 gcd(z', z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0, z' = 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(0, 0), 1 + 0) :|: z1 = 1 + 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(z'' - 1, 0), 1 + 0) :|: z1 = 1 + 0, z' >= 0, z'' - 1 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(z', minus(z1 - 1, 0), 1 + 0) :|: z1 - 1 >= 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 + z1 }-> gcd(z', minus(1 + s'', 1 + min(z'' - 2, z1 - 2)), 1 + (1 + min(z'' - 2, z1 - 2))) :|: s'' >= 0, s'' <= z1 - 2 + (z'' - 2), z' >= 0, z1 - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(0, 0), z'', 1 + 0) :|: z1 = 1 + 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(0, 0), 1 + 0, z1) :|: z1 >= 0, z' = 1 + 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z' - 1, 0), z'', 1 + 0) :|: z1 = 1 + 0, z' - 1 >= 0, z'' >= 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z' - 1, 0), 1 + 0, z1) :|: z1 >= 0, z' - 1 >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z'' - 1, 0), 1 + 0, z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 }-> gcd(minus(z1 - 1, 0), z'', 1 + 0) :|: z1 - 1 >= 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 4 + z'' }-> gcd(minus(1 + s', 1 + min(z' - 2, z'' - 2)), 1 + (1 + min(z' - 2, z'' - 2)), z1) :|: s' >= 0, s' <= z'' - 2 + (z' - 2), z1 >= 0, z' - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 + z1 }-> gcd(minus(1 + s1, 1 + min(z' - 2, z1 - 2)), z'', 1 + (1 + min(z' - 2, z1 - 2))) :|: s1 >= 0, s1 <= z1 - 2 + (z' - 2), z1 - 2 >= 0, z' - 2 >= 0, z'' >= 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'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {min}, {gcd} Previous analysis results are: max: runtime: O(n^1) [2 + z''], size: O(n^1) [z' + z''] minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z'] ---------------------------------------- (59) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: gcd(z', z'', z1) -{ 1 }-> z' :|: z'' = 0, z1 = 0, z' >= 0 gcd(z', z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0, z' = 0 gcd(z', z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0, z' = 0 gcd(z', z'', z1) -{ 5 }-> gcd(s10, z'', 1 + 0) :|: s10 >= 0, s10 <= 0, z1 = 1 + 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s11, z'', 1 + 0) :|: s11 >= 0, s11 <= z1 - 1, z1 - 1 >= 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s3, 1 + 0, z1) :|: s3 >= 0, s3 <= z' - 1, z1 >= 0, z' - 1 >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s4, 1 + 0, z1) :|: s4 >= 0, s4 <= 0, z1 >= 0, z' = 1 + 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s5, 1 + 0, z1) :|: s5 >= 0, s5 <= z'' - 1, z1 >= 0, z'' - 1 >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s9, z'', 1 + 0) :|: s9 >= 0, s9 <= z' - 1, z1 = 1 + 0, z' - 1 >= 0, z'' >= 0 gcd(z', z'', z1) -{ 5 }-> gcd(z', s6, 1 + 0) :|: s6 >= 0, s6 <= z'' - 1, z1 = 1 + 0, z' >= 0, z'' - 1 >= 0 gcd(z', z'', z1) -{ 5 }-> gcd(z', s7, 1 + 0) :|: s7 >= 0, s7 <= 0, z1 = 1 + 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(z', s8, 1 + 0) :|: s8 >= 0, s8 <= z1 - 1, z1 - 1 >= 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 + z1 }-> gcd(z', minus(1 + s'', 1 + min(z'' - 2, z1 - 2)), 1 + (1 + min(z'' - 2, z1 - 2))) :|: s'' >= 0, s'' <= z1 - 2 + (z'' - 2), z' >= 0, z1 - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 + z'' }-> gcd(minus(1 + s', 1 + min(z' - 2, z'' - 2)), 1 + (1 + min(z' - 2, z'' - 2)), z1) :|: s' >= 0, s' <= z'' - 2 + (z' - 2), z1 >= 0, z' - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 + z1 }-> gcd(minus(1 + s1, 1 + min(z' - 2, z1 - 2)), z'', 1 + (1 + min(z' - 2, z1 - 2))) :|: s1 >= 0, s1 <= z1 - 2 + (z' - 2), z1 - 2 >= 0, z' - 2 >= 0, z'' >= 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'' }-> 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 Function symbols to be analyzed: {min}, {gcd} Previous analysis results are: max: runtime: O(n^1) [2 + z''], size: O(n^1) [z' + z''] minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z'] ---------------------------------------- (61) 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'' ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: gcd(z', z'', z1) -{ 1 }-> z' :|: z'' = 0, z1 = 0, z' >= 0 gcd(z', z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0, z' = 0 gcd(z', z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0, z' = 0 gcd(z', z'', z1) -{ 5 }-> gcd(s10, z'', 1 + 0) :|: s10 >= 0, s10 <= 0, z1 = 1 + 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s11, z'', 1 + 0) :|: s11 >= 0, s11 <= z1 - 1, z1 - 1 >= 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s3, 1 + 0, z1) :|: s3 >= 0, s3 <= z' - 1, z1 >= 0, z' - 1 >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s4, 1 + 0, z1) :|: s4 >= 0, s4 <= 0, z1 >= 0, z' = 1 + 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s5, 1 + 0, z1) :|: s5 >= 0, s5 <= z'' - 1, z1 >= 0, z'' - 1 >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s9, z'', 1 + 0) :|: s9 >= 0, s9 <= z' - 1, z1 = 1 + 0, z' - 1 >= 0, z'' >= 0 gcd(z', z'', z1) -{ 5 }-> gcd(z', s6, 1 + 0) :|: s6 >= 0, s6 <= z'' - 1, z1 = 1 + 0, z' >= 0, z'' - 1 >= 0 gcd(z', z'', z1) -{ 5 }-> gcd(z', s7, 1 + 0) :|: s7 >= 0, s7 <= 0, z1 = 1 + 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(z', s8, 1 + 0) :|: s8 >= 0, s8 <= z1 - 1, z1 - 1 >= 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 + z1 }-> gcd(z', minus(1 + s'', 1 + min(z'' - 2, z1 - 2)), 1 + (1 + min(z'' - 2, z1 - 2))) :|: s'' >= 0, s'' <= z1 - 2 + (z'' - 2), z' >= 0, z1 - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 + z'' }-> gcd(minus(1 + s', 1 + min(z' - 2, z'' - 2)), 1 + (1 + min(z' - 2, z'' - 2)), z1) :|: s' >= 0, s' <= z'' - 2 + (z' - 2), z1 >= 0, z' - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 + z1 }-> gcd(minus(1 + s1, 1 + min(z' - 2, z1 - 2)), z'', 1 + (1 + min(z' - 2, z1 - 2))) :|: s1 >= 0, s1 <= z1 - 2 + (z' - 2), z1 - 2 >= 0, z' - 2 >= 0, z'' >= 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'' }-> 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 Function symbols to be analyzed: {min}, {gcd} Previous analysis results are: max: runtime: O(n^1) [2 + z''], size: O(n^1) [z' + z''] minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z'] min: runtime: ?, size: O(n^1) [z''] ---------------------------------------- (63) 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'' ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: gcd(z', z'', z1) -{ 1 }-> z' :|: z'' = 0, z1 = 0, z' >= 0 gcd(z', z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0, z' = 0 gcd(z', z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0, z' = 0 gcd(z', z'', z1) -{ 5 }-> gcd(s10, z'', 1 + 0) :|: s10 >= 0, s10 <= 0, z1 = 1 + 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s11, z'', 1 + 0) :|: s11 >= 0, s11 <= z1 - 1, z1 - 1 >= 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s3, 1 + 0, z1) :|: s3 >= 0, s3 <= z' - 1, z1 >= 0, z' - 1 >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s4, 1 + 0, z1) :|: s4 >= 0, s4 <= 0, z1 >= 0, z' = 1 + 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s5, 1 + 0, z1) :|: s5 >= 0, s5 <= z'' - 1, z1 >= 0, z'' - 1 >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s9, z'', 1 + 0) :|: s9 >= 0, s9 <= z' - 1, z1 = 1 + 0, z' - 1 >= 0, z'' >= 0 gcd(z', z'', z1) -{ 5 }-> gcd(z', s6, 1 + 0) :|: s6 >= 0, s6 <= z'' - 1, z1 = 1 + 0, z' >= 0, z'' - 1 >= 0 gcd(z', z'', z1) -{ 5 }-> gcd(z', s7, 1 + 0) :|: s7 >= 0, s7 <= 0, z1 = 1 + 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(z', s8, 1 + 0) :|: s8 >= 0, s8 <= z1 - 1, z1 - 1 >= 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 4 + z1 }-> gcd(z', minus(1 + s'', 1 + min(z'' - 2, z1 - 2)), 1 + (1 + min(z'' - 2, z1 - 2))) :|: s'' >= 0, s'' <= z1 - 2 + (z'' - 2), z' >= 0, z1 - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 + z'' }-> gcd(minus(1 + s', 1 + min(z' - 2, z'' - 2)), 1 + (1 + min(z' - 2, z'' - 2)), z1) :|: s' >= 0, s' <= z'' - 2 + (z' - 2), z1 >= 0, z' - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 4 + z1 }-> gcd(minus(1 + s1, 1 + min(z' - 2, z1 - 2)), z'', 1 + (1 + min(z' - 2, z1 - 2))) :|: s1 >= 0, s1 <= z1 - 2 + (z' - 2), z1 - 2 >= 0, z' - 2 >= 0, z'' >= 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'' }-> 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 Function symbols to be analyzed: {gcd} Previous analysis results are: max: runtime: O(n^1) [2 + z''], size: O(n^1) [z' + z''] minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z'] min: runtime: O(n^1) [2 + z''], size: O(n^1) [z''] ---------------------------------------- (65) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: gcd(z', z'', z1) -{ 1 }-> z' :|: z'' = 0, z1 = 0, z' >= 0 gcd(z', z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0, z' = 0 gcd(z', z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0, z' = 0 gcd(z', z'', z1) -{ 5 }-> gcd(s10, z'', 1 + 0) :|: s10 >= 0, s10 <= 0, z1 = 1 + 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s11, z'', 1 + 0) :|: s11 >= 0, s11 <= z1 - 1, z1 - 1 >= 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 6 + s13 + 3*z'' }-> gcd(s14, 1 + (1 + s15), z1) :|: s13 >= 0, s13 <= z'' - 2, s14 >= 0, s14 <= 1 + s', s15 >= 0, s15 <= z'' - 2, s' >= 0, s' <= z'' - 2 + (z' - 2), z1 >= 0, z' - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 6 + s19 + 3*z1 }-> gcd(s20, z'', 1 + (1 + s21)) :|: s19 >= 0, s19 <= z1 - 2, s20 >= 0, s20 <= 1 + s1, s21 >= 0, s21 <= z1 - 2, s1 >= 0, s1 <= z1 - 2 + (z' - 2), z1 - 2 >= 0, z' - 2 >= 0, z'' >= 0 gcd(z', z'', z1) -{ 5 }-> gcd(s3, 1 + 0, z1) :|: s3 >= 0, s3 <= z' - 1, z1 >= 0, z' - 1 >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s4, 1 + 0, z1) :|: s4 >= 0, s4 <= 0, z1 >= 0, z' = 1 + 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s5, 1 + 0, z1) :|: s5 >= 0, s5 <= z'' - 1, z1 >= 0, z'' - 1 >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s9, z'', 1 + 0) :|: s9 >= 0, s9 <= z' - 1, z1 = 1 + 0, z' - 1 >= 0, z'' >= 0 gcd(z', z'', z1) -{ 6 + s16 + 3*z1 }-> gcd(z', s17, 1 + (1 + s18)) :|: s16 >= 0, s16 <= z1 - 2, s17 >= 0, s17 <= 1 + s'', s18 >= 0, s18 <= z1 - 2, s'' >= 0, s'' <= z1 - 2 + (z'' - 2), z' >= 0, z1 - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 5 }-> gcd(z', s6, 1 + 0) :|: s6 >= 0, s6 <= z'' - 1, z1 = 1 + 0, z' >= 0, z'' - 1 >= 0 gcd(z', z'', z1) -{ 5 }-> gcd(z', s7, 1 + 0) :|: s7 >= 0, s7 <= 0, z1 = 1 + 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(z', s8, 1 + 0) :|: s8 >= 0, s8 <= z1 - 1, z1 - 1 >= 0, z' >= 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 + s12 :|: s12 >= 0, s12 <= z'' - 1, z' - 1 >= 0, z'' - 1 >= 0 minus(z', z'') -{ 1 + z'' }-> 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 Function symbols to be analyzed: {gcd} Previous analysis results are: max: runtime: O(n^1) [2 + z''], size: O(n^1) [z' + z''] minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z'] min: runtime: O(n^1) [2 + z''], size: O(n^1) [z''] ---------------------------------------- (67) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: gcd after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: gcd(z', z'', z1) -{ 1 }-> z' :|: z'' = 0, z1 = 0, z' >= 0 gcd(z', z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0, z' = 0 gcd(z', z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0, z' = 0 gcd(z', z'', z1) -{ 5 }-> gcd(s10, z'', 1 + 0) :|: s10 >= 0, s10 <= 0, z1 = 1 + 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s11, z'', 1 + 0) :|: s11 >= 0, s11 <= z1 - 1, z1 - 1 >= 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 6 + s13 + 3*z'' }-> gcd(s14, 1 + (1 + s15), z1) :|: s13 >= 0, s13 <= z'' - 2, s14 >= 0, s14 <= 1 + s', s15 >= 0, s15 <= z'' - 2, s' >= 0, s' <= z'' - 2 + (z' - 2), z1 >= 0, z' - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 6 + s19 + 3*z1 }-> gcd(s20, z'', 1 + (1 + s21)) :|: s19 >= 0, s19 <= z1 - 2, s20 >= 0, s20 <= 1 + s1, s21 >= 0, s21 <= z1 - 2, s1 >= 0, s1 <= z1 - 2 + (z' - 2), z1 - 2 >= 0, z' - 2 >= 0, z'' >= 0 gcd(z', z'', z1) -{ 5 }-> gcd(s3, 1 + 0, z1) :|: s3 >= 0, s3 <= z' - 1, z1 >= 0, z' - 1 >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s4, 1 + 0, z1) :|: s4 >= 0, s4 <= 0, z1 >= 0, z' = 1 + 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s5, 1 + 0, z1) :|: s5 >= 0, s5 <= z'' - 1, z1 >= 0, z'' - 1 >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s9, z'', 1 + 0) :|: s9 >= 0, s9 <= z' - 1, z1 = 1 + 0, z' - 1 >= 0, z'' >= 0 gcd(z', z'', z1) -{ 6 + s16 + 3*z1 }-> gcd(z', s17, 1 + (1 + s18)) :|: s16 >= 0, s16 <= z1 - 2, s17 >= 0, s17 <= 1 + s'', s18 >= 0, s18 <= z1 - 2, s'' >= 0, s'' <= z1 - 2 + (z'' - 2), z' >= 0, z1 - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 5 }-> gcd(z', s6, 1 + 0) :|: s6 >= 0, s6 <= z'' - 1, z1 = 1 + 0, z' >= 0, z'' - 1 >= 0 gcd(z', z'', z1) -{ 5 }-> gcd(z', s7, 1 + 0) :|: s7 >= 0, s7 <= 0, z1 = 1 + 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(z', s8, 1 + 0) :|: s8 >= 0, s8 <= z1 - 1, z1 - 1 >= 0, z' >= 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 + s12 :|: s12 >= 0, s12 <= z'' - 1, z' - 1 >= 0, z'' - 1 >= 0 minus(z', z'') -{ 1 + z'' }-> 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 Function symbols to be analyzed: {gcd} Previous analysis results are: max: runtime: O(n^1) [2 + z''], size: O(n^1) [z' + z''] minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z'] min: runtime: O(n^1) [2 + z''], size: O(n^1) [z''] gcd: runtime: ?, size: INF ---------------------------------------- (69) 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: ? ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: gcd(z', z'', z1) -{ 1 }-> z' :|: z'' = 0, z1 = 0, z' >= 0 gcd(z', z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0, z' = 0 gcd(z', z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0, z' = 0 gcd(z', z'', z1) -{ 5 }-> gcd(s10, z'', 1 + 0) :|: s10 >= 0, s10 <= 0, z1 = 1 + 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s11, z'', 1 + 0) :|: s11 >= 0, s11 <= z1 - 1, z1 - 1 >= 0, z'' >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 6 + s13 + 3*z'' }-> gcd(s14, 1 + (1 + s15), z1) :|: s13 >= 0, s13 <= z'' - 2, s14 >= 0, s14 <= 1 + s', s15 >= 0, s15 <= z'' - 2, s' >= 0, s' <= z'' - 2 + (z' - 2), z1 >= 0, z' - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 6 + s19 + 3*z1 }-> gcd(s20, z'', 1 + (1 + s21)) :|: s19 >= 0, s19 <= z1 - 2, s20 >= 0, s20 <= 1 + s1, s21 >= 0, s21 <= z1 - 2, s1 >= 0, s1 <= z1 - 2 + (z' - 2), z1 - 2 >= 0, z' - 2 >= 0, z'' >= 0 gcd(z', z'', z1) -{ 5 }-> gcd(s3, 1 + 0, z1) :|: s3 >= 0, s3 <= z' - 1, z1 >= 0, z' - 1 >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s4, 1 + 0, z1) :|: s4 >= 0, s4 <= 0, z1 >= 0, z' = 1 + 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s5, 1 + 0, z1) :|: s5 >= 0, s5 <= z'' - 1, z1 >= 0, z'' - 1 >= 0, z' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(s9, z'', 1 + 0) :|: s9 >= 0, s9 <= z' - 1, z1 = 1 + 0, z' - 1 >= 0, z'' >= 0 gcd(z', z'', z1) -{ 6 + s16 + 3*z1 }-> gcd(z', s17, 1 + (1 + s18)) :|: s16 >= 0, s16 <= z1 - 2, s17 >= 0, s17 <= 1 + s'', s18 >= 0, s18 <= z1 - 2, s'' >= 0, s'' <= z1 - 2 + (z'' - 2), z' >= 0, z1 - 2 >= 0, z'' - 2 >= 0 gcd(z', z'', z1) -{ 5 }-> gcd(z', s6, 1 + 0) :|: s6 >= 0, s6 <= z'' - 1, z1 = 1 + 0, z' >= 0, z'' - 1 >= 0 gcd(z', z'', z1) -{ 5 }-> gcd(z', s7, 1 + 0) :|: s7 >= 0, s7 <= 0, z1 = 1 + 0, z' >= 0, z'' = 1 + 0 gcd(z', z'', z1) -{ 5 }-> gcd(z', s8, 1 + 0) :|: s8 >= 0, s8 <= z1 - 1, z1 - 1 >= 0, z' >= 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 + s12 :|: s12 >= 0, s12 <= z'' - 1, z' - 1 >= 0, z'' - 1 >= 0 minus(z', z'') -{ 1 + z'' }-> 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 Function symbols to be analyzed: {gcd} Previous analysis results are: max: runtime: O(n^1) [2 + z''], size: O(n^1) [z' + z''] minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z'] min: runtime: O(n^1) [2 + z''], size: O(n^1) [z''] gcd: runtime: INF, size: INF ---------------------------------------- (71) 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] min(v0, v1) -> null_min [0] max(v0, v1) -> null_max [0] gcd(v0, v1, v2) -> null_gcd [0] And the following fresh constants: null_minus, null_min, null_max, null_gcd ---------------------------------------- (72) 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)) -> minus(x, y) [1] gcd(s(x), s(y), z) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)), z) [1] gcd(x, s(y), s(z)) -> gcd(x, minus(max(y, z), min(y, z)), s(min(y, z))) [1] gcd(s(x), y, s(z)) -> gcd(minus(max(x, z), min(x, z)), y, s(min(x, z))) [1] gcd(x, 0, 0) -> x [1] gcd(0, y, 0) -> y [1] gcd(0, 0, z) -> z [1] minus(v0, v1) -> null_minus [0] min(v0, v1) -> null_min [0] max(v0, v1) -> null_max [0] gcd(v0, v1, v2) -> null_gcd [0] The TRS has the following type information: min :: 0:s:null_minus:null_min:null_max:null_gcd -> 0:s:null_minus:null_min:null_max:null_gcd -> 0:s:null_minus:null_min:null_max:null_gcd 0 :: 0:s:null_minus:null_min:null_max:null_gcd s :: 0:s:null_minus:null_min:null_max:null_gcd -> 0:s:null_minus:null_min:null_max:null_gcd max :: 0:s:null_minus:null_min:null_max:null_gcd -> 0:s:null_minus:null_min:null_max:null_gcd -> 0:s:null_minus:null_min:null_max:null_gcd minus :: 0:s:null_minus:null_min:null_max:null_gcd -> 0:s:null_minus:null_min:null_max:null_gcd -> 0:s:null_minus:null_min:null_max:null_gcd gcd :: 0:s:null_minus:null_min:null_max:null_gcd -> 0:s:null_minus:null_min:null_max:null_gcd -> 0:s:null_minus:null_min:null_max:null_gcd -> 0:s:null_minus:null_min:null_max:null_gcd null_minus :: 0:s:null_minus:null_min:null_max:null_gcd null_min :: 0:s:null_minus:null_min:null_max:null_gcd null_max :: 0:s:null_minus:null_min:null_max:null_gcd null_gcd :: 0:s:null_minus:null_min:null_max:null_gcd Rewrite Strategy: INNERMOST ---------------------------------------- (73) 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_min => 0 null_max => 0 null_gcd => 0 ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: gcd(z', z'', z1) -{ 1 }-> x :|: z'' = 0, z1 = 0, z' = x, x >= 0 gcd(z', z'', z1) -{ 1 }-> y :|: z1 = 0, z'' = y, y >= 0, z' = 0 gcd(z', z'', z1) -{ 1 }-> z :|: z'' = 0, z1 = z, z >= 0, z' = 0 gcd(z', z'', z1) -{ 1 }-> gcd(x, minus(max(y, z), min(y, z)), 1 + min(y, z)) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y, z1 = 1 + z gcd(z', z'', z1) -{ 1 }-> gcd(minus(max(x, y), min(x, y)), 1 + min(x, y), z) :|: z' = 1 + x, z1 = z, z >= 0, x >= 0, y >= 0, z'' = 1 + y gcd(z', z'', z1) -{ 1 }-> gcd(minus(max(x, z), min(x, z)), y, 1 + min(x, z)) :|: z' = 1 + x, z >= 0, z'' = y, x >= 0, y >= 0, z1 = 1 + z gcd(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0 max(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0 max(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0 max(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 max(z', z'') -{ 1 }-> 1 + max(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y min(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = x, x >= 0 min(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0 min(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 min(z', z'') -{ 1 }-> 1 + min(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y minus(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0 minus(z', z'') -{ 1 }-> minus(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y minus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (75) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (76) 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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2) gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))) gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))) gcd(z0, 0, 0) -> z0 gcd(0, z0, 0) -> z0 gcd(0, 0, 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)) -'(z0, 0) -> c6 -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(z0, 0, 0) -> c17 GCD(0, z0, 0) -> c18 GCD(0, 0, z0) -> c19 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)) -'(z0, 0) -> c6 -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(z0, 0, 0) -> c17 GCD(0, z0, 0) -> c18 GCD(0, 0, z0) -> c19 K tuples:none Defined Rule Symbols: min_2, max_2, -_2, gcd_3 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c, c1, c2_1, c3, c4, c5_1, c6, c7_1, c8_3, c9_3, c10_2, c11_3, c12_3, c13_2, c14_3, c15_3, c16_2, c17, c18, c19 ---------------------------------------- (77) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 8 trailing nodes: GCD(0, 0, z0) -> c19 GCD(0, z0, 0) -> c18 -'(z0, 0) -> c6 GCD(z0, 0, 0) -> c17 MAX(z0, 0) -> c3 MAX(0, z0) -> c4 MIN(0, z0) -> c1 MIN(z0, 0) -> c ---------------------------------------- (78) 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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2) gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))) gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))) gcd(z0, 0, 0) -> z0 gcd(0, z0, 0) -> z0 gcd(0, 0, z0) -> z0 Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) K tuples:none Defined Rule Symbols: min_2, max_2, -_2, gcd_3 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c8_3, c9_3, c10_2, c11_3, c12_3, c13_2, c14_3, c15_3, c16_2 ---------------------------------------- (79) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2) gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))) gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))) gcd(z0, 0, 0) -> z0 gcd(0, z0, 0) -> z0 gcd(0, 0, z0) -> z0 ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c8_3, c9_3, c10_2, c11_3, c12_3, c13_2, c14_3, c15_3, c16_2 ---------------------------------------- (81) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (82) 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) The (relative) TRS S consists of the following rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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 ---------------------------------------- (83) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (84) 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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 ---------------------------------------- (85) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (86) 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] -'(s(z0), s(z1)) -> c7(-'(z0, z1)) [1] GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) [1] GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) [1] GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) [1] GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) [1] GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) [1] GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) [1] GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) [1] GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) [1] GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) [1] -(z0, 0) -> z0 [0] -(s(z0), s(z1)) -> -(z0, 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 ---------------------------------------- (87) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: - => minus ---------------------------------------- (88) 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] -'(s(z0), s(z1)) -> c7(-'(z0, z1)) [1] GCD(s(z0), s(z1), z2) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) [1] GCD(s(z0), s(z1), z2) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) [1] GCD(s(z0), s(z1), z2) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) [1] GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, minus(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) [1] GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, minus(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) [1] GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, minus(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) [1] GCD(s(z0), z1, s(z2)) -> c14(GCD(minus(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) [1] GCD(s(z0), z1, s(z2)) -> c15(GCD(minus(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) [1] GCD(s(z0), z1, s(z2)) -> c16(GCD(minus(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) [1] minus(z0, 0) -> z0 [0] minus(s(z0), s(z1)) -> minus(z0, 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 ---------------------------------------- (89) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (90) 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] -'(s(z0), s(z1)) -> c7(-'(z0, z1)) [1] GCD(s(z0), s(z1), z2) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) [1] GCD(s(z0), s(z1), z2) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) [1] GCD(s(z0), s(z1), z2) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) [1] GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, minus(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) [1] GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, minus(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) [1] GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, minus(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) [1] GCD(s(z0), z1, s(z2)) -> c14(GCD(minus(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) [1] GCD(s(z0), z1, s(z2)) -> c15(GCD(minus(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) [1] GCD(s(z0), z1, s(z2)) -> c16(GCD(minus(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) [1] minus(z0, 0) -> z0 [0] minus(s(z0), s(z1)) -> minus(z0, 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 -' :: s:0 -> s:0 -> c7 c7 :: c7 -> c7 GCD :: s:0 -> s:0 -> s:0 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c8 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c7 -> c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 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:c11:c12:c13:c14:c15:c16 -> c7 -> c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c10 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c11 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c7 -> c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c12 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c7 -> c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c13 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c14 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c7 -> c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c15 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c7 -> c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c16 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 0 :: s:0 Rewrite Strategy: INNERMOST ---------------------------------------- (91) 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 -'_2 GCD_3 (c) The following functions are completely defined: minus_2 max_2 min_2 Due to the following rules being added: minus(v0, v1) -> 0 [0] max(v0, v1) -> 0 [0] min(v0, v1) -> 0 [0] And the following fresh constants: const, const1, const2, const3 ---------------------------------------- (92) 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] -'(s(z0), s(z1)) -> c7(-'(z0, z1)) [1] GCD(s(z0), s(z1), z2) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) [1] GCD(s(z0), s(z1), z2) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) [1] GCD(s(z0), s(z1), z2) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) [1] GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, minus(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) [1] GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, minus(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) [1] GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, minus(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) [1] GCD(s(z0), z1, s(z2)) -> c14(GCD(minus(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) [1] GCD(s(z0), z1, s(z2)) -> c15(GCD(minus(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) [1] GCD(s(z0), z1, s(z2)) -> c16(GCD(minus(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) [1] minus(z0, 0) -> z0 [0] minus(s(z0), s(z1)) -> minus(z0, 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] 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 -' :: s:0 -> s:0 -> c7 c7 :: c7 -> c7 GCD :: s:0 -> s:0 -> s:0 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c8 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c7 -> c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 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:c11:c12:c13:c14:c15:c16 -> c7 -> c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c10 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c11 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c7 -> c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c12 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c7 -> c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c13 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c14 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c7 -> c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c15 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c7 -> c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c16 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 0 :: s:0 const :: c2 const1 :: c5 const2 :: c7 const3 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 Rewrite Strategy: INNERMOST ---------------------------------------- (93) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (94) 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] -'(s(z0), s(z1)) -> c7(-'(z0, z1)) [1] GCD(s(z0), s(z1), z2) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) [1] GCD(s(z0), s(z1), z2) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) [1] GCD(s(z0), s(z1), z2) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) [1] GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, minus(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) [1] GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, minus(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) [1] GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, minus(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) [1] GCD(s(z0), z1, s(z2)) -> c14(GCD(minus(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) [1] GCD(s(z0), z1, s(z2)) -> c15(GCD(minus(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) [1] GCD(s(z0), z1, s(z2)) -> c16(GCD(minus(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) [1] minus(z0, 0) -> z0 [0] minus(s(z0), s(z1)) -> minus(z0, 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] 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 -' :: s:0 -> s:0 -> c7 c7 :: c7 -> c7 GCD :: s:0 -> s:0 -> s:0 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c8 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c7 -> c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 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:c11:c12:c13:c14:c15:c16 -> c7 -> c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c10 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c11 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c7 -> c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c12 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c7 -> c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c13 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c14 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c7 -> c5 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c15 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c7 -> c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 c16 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 -> c2 -> c8:c9:c10:c11:c12:c13:c14:c15:c16 0 :: s:0 const :: c2 const1 :: c5 const2 :: c7 const3 :: c8:c9:c10:c11:c12:c13:c14:c15:c16 Rewrite Strategy: INNERMOST ---------------------------------------- (95) 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 ---------------------------------------- (96) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ 1 }-> 1 + -'(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z0, minus(max(z1, z2), min(z1, z2)), 1 + min(z1, z2)) + MIN(z1, z2) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1, z'' = 1 + z2, z2 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z0, z1), min(z0, z1)), 1 + min(z0, z1), z2) + MIN(z0, z1) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z2 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z0, z2), min(z0, z2)), z1, 1 + min(z0, z2)) + MIN(z0, z2) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0, z'' = 1 + z2, z2 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z0, minus(max(z1, z2), min(z1, z2)), 1 + min(z1, z2)) + -'(max(z1, z2), min(z1, z2)) + MIN(z1, z2) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1, z'' = 1 + z2, z2 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z0, minus(max(z1, z2), min(z1, z2)), 1 + min(z1, z2)) + -'(max(z1, z2), min(z1, z2)) + MAX(z1, z2) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1, z'' = 1 + z2, z2 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z0, z1), min(z0, z1)), 1 + min(z0, z1), z2) + -'(max(z0, z1), min(z0, z1)) + MIN(z0, z1) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z2 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z0, z1), min(z0, z1)), 1 + min(z0, z1), z2) + -'(max(z0, z1), min(z0, z1)) + MAX(z0, z1) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z2 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z0, z2), min(z0, z2)), z1, 1 + min(z0, z2)) + -'(max(z0, z2), min(z0, z2)) + MIN(z0, z2) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0, z'' = 1 + z2, z2 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z0, z2), min(z0, z2)), z1, 1 + min(z0, z2)) + -'(max(z0, z2), min(z0, z2)) + MAX(z0, z2) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0, z'' = 1 + z2, z2 >= 0 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 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 }-> minus(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 ---------------------------------------- (97) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (98) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, 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 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + MIN(z - 1, z'' - 1) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)) + MIN(z' - 1, z'' - 1) :|: z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)) + MAX(z' - 1, z'' - 1) :|: z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(max(z - 1, z'' - 1), min(z - 1, z'' - 1)) + MIN(z - 1, z'' - 1) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(max(z - 1, z'' - 1), min(z - 1, z'' - 1)) + MAX(z - 1, z'' - 1) :|: z' >= 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') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, 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 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ---------------------------------------- (99) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { MIN } { max } { -' } { minus } { min } { MAX } { GCD } ---------------------------------------- (100) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, 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 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + MIN(z - 1, z'' - 1) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)) + MIN(z' - 1, z'' - 1) :|: z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)) + MAX(z' - 1, z'' - 1) :|: z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(max(z - 1, z'' - 1), min(z - 1, z'' - 1)) + MIN(z - 1, z'' - 1) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(max(z - 1, z'' - 1), min(z - 1, z'' - 1)) + MAX(z - 1, z'' - 1) :|: z' >= 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') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, 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 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {MIN}, {max}, {-'}, {minus}, {min}, {MAX}, {GCD} ---------------------------------------- (101) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (102) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, 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 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + MIN(z - 1, z'' - 1) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)) + MIN(z' - 1, z'' - 1) :|: z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)) + MAX(z' - 1, z'' - 1) :|: z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(max(z - 1, z'' - 1), min(z - 1, z'' - 1)) + MIN(z - 1, z'' - 1) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(max(z - 1, z'' - 1), min(z - 1, z'' - 1)) + MAX(z - 1, z'' - 1) :|: z' >= 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') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, 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 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {MIN}, {max}, {-'}, {minus}, {min}, {MAX}, {GCD} ---------------------------------------- (103) 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 ---------------------------------------- (104) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, 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 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + MIN(z - 1, z'' - 1) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)) + MIN(z' - 1, z'' - 1) :|: z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)) + MAX(z' - 1, z'' - 1) :|: z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(max(z - 1, z'' - 1), min(z - 1, z'' - 1)) + MIN(z - 1, z'' - 1) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(max(z - 1, z'' - 1), min(z - 1, z'' - 1)) + MAX(z - 1, z'' - 1) :|: z' >= 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') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, 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 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {MIN}, {max}, {-'}, {minus}, {min}, {MAX}, {GCD} Previous analysis results are: MIN: runtime: ?, size: O(1) [0] ---------------------------------------- (105) 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' ---------------------------------------- (106) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, 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 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + MIN(z - 1, z'' - 1) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)) + MIN(z' - 1, z'' - 1) :|: z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)) + MAX(z' - 1, z'' - 1) :|: z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(max(z - 1, z'' - 1), min(z - 1, z'' - 1)) + MIN(z - 1, z'' - 1) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(max(z - 1, z'' - 1), min(z - 1, z'' - 1)) + MAX(z - 1, z'' - 1) :|: z' >= 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') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, 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 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {max}, {-'}, {minus}, {min}, {MAX}, {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] ---------------------------------------- (107) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (108) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)) + MAX(z' - 1, z'' - 1) :|: z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(max(z - 1, z' - 1), min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(max(z - 1, z'' - 1), min(z - 1, z'' - 1)) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(max(z - 1, z'' - 1), min(z - 1, z'' - 1)) + MAX(z - 1, z'' - 1) :|: z' >= 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 + s :|: s >= 0, s <= 0, z' - 1 >= 0, 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 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {max}, {-'}, {minus}, {min}, {MAX}, {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] ---------------------------------------- (109) 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' ---------------------------------------- (110) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)) + MAX(z' - 1, z'' - 1) :|: z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(max(z - 1, z' - 1), min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(max(z - 1, z'' - 1), min(z - 1, z'' - 1)) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(max(z - 1, z'' - 1), min(z - 1, z'' - 1)) + MAX(z - 1, z'' - 1) :|: z' >= 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 + s :|: s >= 0, s <= 0, z' - 1 >= 0, 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 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {max}, {-'}, {minus}, {min}, {MAX}, {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (111) 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 ---------------------------------------- (112) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(max(z' - 1, z'' - 1), min(z' - 1, z'' - 1)) + MAX(z' - 1, z'' - 1) :|: z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(max(z - 1, z' - 1), min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z' - 1), min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(max(z - 1, z' - 1), min(z - 1, z' - 1)) + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(max(z - 1, z'' - 1), min(z - 1, z'' - 1)) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z - 1, z'' - 1), min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(max(z - 1, z'' - 1), min(z - 1, z'' - 1)) + MAX(z - 1, z'' - 1) :|: z' >= 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 + s :|: s >= 0, s <= 0, z' - 1 >= 0, 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 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {-'}, {minus}, {min}, {MAX}, {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] ---------------------------------------- (113) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (114) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(s14, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + s2 :|: s14 >= 0, s14 <= z'' - 1 + (z' - 1), s2 >= 0, s2 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(s19, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + s4 :|: s19 >= 0, s19 <= z'' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(s9, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + s'' :|: s9 >= 0, s9 <= z' - 1 + (z - 1), s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(s10, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(s11, min(z' - 1, z'' - 1)) + MAX(z' - 1, z'' - 1) :|: s10 >= 0, s10 <= z'' - 1 + (z' - 1), s11 >= 0, s11 <= z'' - 1 + (z' - 1), z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(s12, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(s13, min(z' - 1, z'' - 1)) + s1 :|: s12 >= 0, s12 <= z'' - 1 + (z' - 1), s13 >= 0, s13 <= z'' - 1 + (z' - 1), s1 >= 0, s1 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(s15, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(s16, min(z - 1, z'' - 1)) + MAX(z - 1, z'' - 1) :|: s15 >= 0, s15 <= z'' - 1 + (z - 1), s16 >= 0, s16 <= z'' - 1 + (z - 1), z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(s17, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(s18, min(z - 1, z'' - 1)) + s3 :|: s17 >= 0, s17 <= z'' - 1 + (z - 1), s18 >= 0, s18 <= z'' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(s5, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(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, z'' >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(s7, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(s8, min(z - 1, z' - 1)) + s' :|: s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, 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 + s20 :|: s20 >= 0, s20 <= 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 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {-'}, {minus}, {min}, {MAX}, {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] ---------------------------------------- (115) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: -' after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (116) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(s14, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + s2 :|: s14 >= 0, s14 <= z'' - 1 + (z' - 1), s2 >= 0, s2 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(s19, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + s4 :|: s19 >= 0, s19 <= z'' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(s9, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + s'' :|: s9 >= 0, s9 <= z' - 1 + (z - 1), s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(s10, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(s11, min(z' - 1, z'' - 1)) + MAX(z' - 1, z'' - 1) :|: s10 >= 0, s10 <= z'' - 1 + (z' - 1), s11 >= 0, s11 <= z'' - 1 + (z' - 1), z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(s12, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(s13, min(z' - 1, z'' - 1)) + s1 :|: s12 >= 0, s12 <= z'' - 1 + (z' - 1), s13 >= 0, s13 <= z'' - 1 + (z' - 1), s1 >= 0, s1 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(s15, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(s16, min(z - 1, z'' - 1)) + MAX(z - 1, z'' - 1) :|: s15 >= 0, s15 <= z'' - 1 + (z - 1), s16 >= 0, s16 <= z'' - 1 + (z - 1), z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(s17, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(s18, min(z - 1, z'' - 1)) + s3 :|: s17 >= 0, s17 <= z'' - 1 + (z - 1), s18 >= 0, s18 <= z'' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(s5, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(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, z'' >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(s7, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(s8, min(z - 1, z' - 1)) + s' :|: s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, 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 + s20 :|: s20 >= 0, s20 <= 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 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {-'}, {minus}, {min}, {MAX}, {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] -': runtime: ?, size: O(1) [0] ---------------------------------------- (117) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: -' after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (118) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(s14, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + s2 :|: s14 >= 0, s14 <= z'' - 1 + (z' - 1), s2 >= 0, s2 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(s19, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + s4 :|: s19 >= 0, s19 <= z'' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(s9, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + s'' :|: s9 >= 0, s9 <= z' - 1 + (z - 1), s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(s10, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(s11, min(z' - 1, z'' - 1)) + MAX(z' - 1, z'' - 1) :|: s10 >= 0, s10 <= z'' - 1 + (z' - 1), s11 >= 0, s11 <= z'' - 1 + (z' - 1), z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(s12, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(s13, min(z' - 1, z'' - 1)) + s1 :|: s12 >= 0, s12 <= z'' - 1 + (z' - 1), s13 >= 0, s13 <= z'' - 1 + (z' - 1), s1 >= 0, s1 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(s15, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(s16, min(z - 1, z'' - 1)) + MAX(z - 1, z'' - 1) :|: s15 >= 0, s15 <= z'' - 1 + (z - 1), s16 >= 0, s16 <= z'' - 1 + (z - 1), z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(s17, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(s18, min(z - 1, z'' - 1)) + s3 :|: s17 >= 0, s17 <= z'' - 1 + (z - 1), s18 >= 0, s18 <= z'' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(s5, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(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, z'' >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(s7, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(s8, min(z - 1, z' - 1)) + s' :|: s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, 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 + s20 :|: s20 >= 0, s20 <= 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 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {minus}, {min}, {MAX}, {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] -': runtime: O(n^1) [z'], size: O(1) [0] ---------------------------------------- (119) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (120) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(s14, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + s2 :|: s14 >= 0, s14 <= z'' - 1 + (z' - 1), s2 >= 0, s2 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(s19, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + s4 :|: s19 >= 0, s19 <= z'' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(s9, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + s'' :|: s9 >= 0, s9 <= z' - 1 + (z - 1), s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(s10, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(s11, min(z' - 1, z'' - 1)) + MAX(z' - 1, z'' - 1) :|: s10 >= 0, s10 <= z'' - 1 + (z' - 1), s11 >= 0, s11 <= z'' - 1 + (z' - 1), z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(s12, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(s13, min(z' - 1, z'' - 1)) + s1 :|: s12 >= 0, s12 <= z'' - 1 + (z' - 1), s13 >= 0, s13 <= z'' - 1 + (z' - 1), s1 >= 0, s1 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(s15, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(s16, min(z - 1, z'' - 1)) + MAX(z - 1, z'' - 1) :|: s15 >= 0, s15 <= z'' - 1 + (z - 1), s16 >= 0, s16 <= z'' - 1 + (z - 1), z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(s17, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(s18, min(z - 1, z'' - 1)) + s3 :|: s17 >= 0, s17 <= z'' - 1 + (z - 1), s18 >= 0, s18 <= z'' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(s5, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(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, z'' >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(s7, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(s8, min(z - 1, z' - 1)) + s' :|: s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, 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 + s20 :|: s20 >= 0, s20 <= 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 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {minus}, {min}, {MAX}, {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] -': runtime: O(n^1) [z'], size: O(1) [0] ---------------------------------------- (121) 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 ---------------------------------------- (122) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(s14, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + s2 :|: s14 >= 0, s14 <= z'' - 1 + (z' - 1), s2 >= 0, s2 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(s19, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + s4 :|: s19 >= 0, s19 <= z'' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(s9, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + s'' :|: s9 >= 0, s9 <= z' - 1 + (z - 1), s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(s10, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(s11, min(z' - 1, z'' - 1)) + MAX(z' - 1, z'' - 1) :|: s10 >= 0, s10 <= z'' - 1 + (z' - 1), s11 >= 0, s11 <= z'' - 1 + (z' - 1), z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(s12, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(s13, min(z' - 1, z'' - 1)) + s1 :|: s12 >= 0, s12 <= z'' - 1 + (z' - 1), s13 >= 0, s13 <= z'' - 1 + (z' - 1), s1 >= 0, s1 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(s15, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(s16, min(z - 1, z'' - 1)) + MAX(z - 1, z'' - 1) :|: s15 >= 0, s15 <= z'' - 1 + (z - 1), s16 >= 0, s16 <= z'' - 1 + (z - 1), z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(s17, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(s18, min(z - 1, z'' - 1)) + s3 :|: s17 >= 0, s17 <= z'' - 1 + (z - 1), s18 >= 0, s18 <= z'' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(s5, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(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, z'' >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(s7, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(s8, min(z - 1, z' - 1)) + s' :|: s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, 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 + s20 :|: s20 >= 0, s20 <= 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 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {minus}, {min}, {MAX}, {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] -': runtime: O(n^1) [z'], size: O(1) [0] minus: runtime: ?, size: O(n^1) [z] ---------------------------------------- (123) 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 ---------------------------------------- (124) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(s14, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + s2 :|: s14 >= 0, s14 <= z'' - 1 + (z' - 1), s2 >= 0, s2 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(s19, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + s4 :|: s19 >= 0, s19 <= z'' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(s9, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + s'' :|: s9 >= 0, s9 <= z' - 1 + (z - 1), s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(s10, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(s11, min(z' - 1, z'' - 1)) + MAX(z' - 1, z'' - 1) :|: s10 >= 0, s10 <= z'' - 1 + (z' - 1), s11 >= 0, s11 <= z'' - 1 + (z' - 1), z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(s12, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(s13, min(z' - 1, z'' - 1)) + s1 :|: s12 >= 0, s12 <= z'' - 1 + (z' - 1), s13 >= 0, s13 <= z'' - 1 + (z' - 1), s1 >= 0, s1 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(s15, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(s16, min(z - 1, z'' - 1)) + MAX(z - 1, z'' - 1) :|: s15 >= 0, s15 <= z'' - 1 + (z - 1), s16 >= 0, s16 <= z'' - 1 + (z - 1), z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(s17, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(s18, min(z - 1, z'' - 1)) + s3 :|: s17 >= 0, s17 <= z'' - 1 + (z - 1), s18 >= 0, s18 <= z'' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(s5, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(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, z'' >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(s7, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(s8, min(z - 1, z' - 1)) + s' :|: s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, 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 + s20 :|: s20 >= 0, s20 <= 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 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {min}, {MAX}, {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] -': runtime: O(n^1) [z'], size: O(1) [0] minus: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (125) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (126) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(s14, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + s2 :|: s14 >= 0, s14 <= z'' - 1 + (z' - 1), s2 >= 0, s2 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(s19, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + s4 :|: s19 >= 0, s19 <= z'' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(s9, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + s'' :|: s9 >= 0, s9 <= z' - 1 + (z - 1), s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(s10, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(s11, min(z' - 1, z'' - 1)) + MAX(z' - 1, z'' - 1) :|: s10 >= 0, s10 <= z'' - 1 + (z' - 1), s11 >= 0, s11 <= z'' - 1 + (z' - 1), z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(s12, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(s13, min(z' - 1, z'' - 1)) + s1 :|: s12 >= 0, s12 <= z'' - 1 + (z' - 1), s13 >= 0, s13 <= z'' - 1 + (z' - 1), s1 >= 0, s1 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(s15, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(s16, min(z - 1, z'' - 1)) + MAX(z - 1, z'' - 1) :|: s15 >= 0, s15 <= z'' - 1 + (z - 1), s16 >= 0, s16 <= z'' - 1 + (z - 1), z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(s17, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(s18, min(z - 1, z'' - 1)) + s3 :|: s17 >= 0, s17 <= z'' - 1 + (z - 1), s18 >= 0, s18 <= z'' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(s5, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(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, z'' >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(s7, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(s8, min(z - 1, z' - 1)) + s' :|: s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, 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 + s20 :|: s20 >= 0, s20 <= 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 }-> 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 Function symbols to be analyzed: {min}, {MAX}, {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] -': runtime: O(n^1) [z'], size: O(1) [0] minus: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (127) 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' ---------------------------------------- (128) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(s14, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + s2 :|: s14 >= 0, s14 <= z'' - 1 + (z' - 1), s2 >= 0, s2 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(s19, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + s4 :|: s19 >= 0, s19 <= z'' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(s9, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + s'' :|: s9 >= 0, s9 <= z' - 1 + (z - 1), s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(s10, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(s11, min(z' - 1, z'' - 1)) + MAX(z' - 1, z'' - 1) :|: s10 >= 0, s10 <= z'' - 1 + (z' - 1), s11 >= 0, s11 <= z'' - 1 + (z' - 1), z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(s12, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(s13, min(z' - 1, z'' - 1)) + s1 :|: s12 >= 0, s12 <= z'' - 1 + (z' - 1), s13 >= 0, s13 <= z'' - 1 + (z' - 1), s1 >= 0, s1 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(s15, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(s16, min(z - 1, z'' - 1)) + MAX(z - 1, z'' - 1) :|: s15 >= 0, s15 <= z'' - 1 + (z - 1), s16 >= 0, s16 <= z'' - 1 + (z - 1), z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(s17, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(s18, min(z - 1, z'' - 1)) + s3 :|: s17 >= 0, s17 <= z'' - 1 + (z - 1), s18 >= 0, s18 <= z'' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(s5, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(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, z'' >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(s7, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(s8, min(z - 1, z' - 1)) + s' :|: s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, 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 + s20 :|: s20 >= 0, s20 <= 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 }-> 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 Function symbols to be analyzed: {min}, {MAX}, {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] -': runtime: O(n^1) [z'], size: O(1) [0] minus: runtime: O(1) [0], size: O(n^1) [z] min: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (129) 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 ---------------------------------------- (130) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(s14, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + s2 :|: s14 >= 0, s14 <= z'' - 1 + (z' - 1), s2 >= 0, s2 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(s19, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + s4 :|: s19 >= 0, s19 <= z'' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(s9, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + s'' :|: s9 >= 0, s9 <= z' - 1 + (z - 1), s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z, minus(s10, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(s11, min(z' - 1, z'' - 1)) + MAX(z' - 1, z'' - 1) :|: s10 >= 0, s10 <= z'' - 1 + (z' - 1), s11 >= 0, s11 <= z'' - 1 + (z' - 1), z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, minus(s12, min(z' - 1, z'' - 1)), 1 + min(z' - 1, z'' - 1)) + -'(s13, min(z' - 1, z'' - 1)) + s1 :|: s12 >= 0, s12 <= z'' - 1 + (z' - 1), s13 >= 0, s13 <= z'' - 1 + (z' - 1), s1 >= 0, s1 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(s15, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(s16, min(z - 1, z'' - 1)) + MAX(z - 1, z'' - 1) :|: s15 >= 0, s15 <= z'' - 1 + (z - 1), s16 >= 0, s16 <= z'' - 1 + (z - 1), z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(minus(s17, min(z - 1, z'' - 1)), z', 1 + min(z - 1, z'' - 1)) + -'(s18, min(z - 1, z'' - 1)) + s3 :|: s17 >= 0, s17 <= z'' - 1 + (z - 1), s18 >= 0, s18 <= z'' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(s5, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(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, z'' >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(minus(s7, min(z - 1, z' - 1)), 1 + min(z - 1, z' - 1), z'') + -'(s8, min(z - 1, z' - 1)) + s' :|: s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, 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 + s20 :|: s20 >= 0, s20 <= 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 }-> 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 Function symbols to be analyzed: {MAX}, {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] -': runtime: O(n^1) [z'], size: O(1) [0] minus: runtime: O(1) [0], size: O(n^1) [z] min: runtime: O(1) [0], size: O(n^1) [z'] ---------------------------------------- (131) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (132) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(s34, 1 + s35, z'') + s'' :|: s33 >= 0, s33 <= z' - 1, s34 >= 0, s34 <= s9, s35 >= 0, s35 <= z' - 1, s9 >= 0, s9 <= z' - 1 + (z - 1), s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(s60, z', 1 + s61) + s4 :|: s59 >= 0, s59 <= z'' - 1, s60 >= 0, s60 <= s19, s61 >= 0, s61 <= z'' - 1, s19 >= 0, s19 <= z'' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, s47, 1 + s48) + s2 :|: s46 >= 0, s46 <= z'' - 1, s47 >= 0, s47 <= s14, s48 >= 0, s48 <= z'' - 1, s14 >= 0, s14 <= z'' - 1 + (z' - 1), s2 >= 0, s2 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 + s26 }-> 1 + GCD(s24, 1 + s25, z'') + s27 + MAX(z - 1, z' - 1) :|: s23 >= 0, s23 <= z' - 1, s24 >= 0, s24 <= s5, s25 >= 0, s25 <= z' - 1, s26 >= 0, s26 <= z' - 1, s27 >= 0, s27 <= 0, s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ s31 + z' }-> 1 + GCD(s29, 1 + s30, z'') + s32 + s' :|: s28 >= 0, s28 <= z' - 1, s29 >= 0, s29 <= s7, s30 >= 0, s30 <= z' - 1, s31 >= 0, s31 <= z' - 1, s32 >= 0, s32 <= 0, s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 + s52 }-> 1 + GCD(s50, z', 1 + s51) + s53 + MAX(z - 1, z'' - 1) :|: s49 >= 0, s49 <= z'' - 1, s50 >= 0, s50 <= s15, s51 >= 0, s51 <= z'' - 1, s52 >= 0, s52 <= z'' - 1, s53 >= 0, s53 <= 0, s15 >= 0, s15 <= z'' - 1 + (z - 1), s16 >= 0, s16 <= z'' - 1 + (z - 1), z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ s57 + z'' }-> 1 + GCD(s55, z', 1 + s56) + s58 + s3 :|: s54 >= 0, s54 <= z'' - 1, s55 >= 0, s55 <= s17, s56 >= 0, s56 <= z'' - 1, s57 >= 0, s57 <= z'' - 1, s58 >= 0, s58 <= 0, s17 >= 0, s17 <= z'' - 1 + (z - 1), s18 >= 0, s18 <= z'' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 + s39 }-> 1 + GCD(z, s37, 1 + s38) + s40 + MAX(z' - 1, z'' - 1) :|: s36 >= 0, s36 <= z'' - 1, s37 >= 0, s37 <= s10, s38 >= 0, s38 <= z'' - 1, s39 >= 0, s39 <= z'' - 1, s40 >= 0, s40 <= 0, s10 >= 0, s10 <= z'' - 1 + (z' - 1), s11 >= 0, s11 <= z'' - 1 + (z' - 1), z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ s44 + z'' }-> 1 + GCD(z, s42, 1 + s43) + s45 + s1 :|: s41 >= 0, s41 <= z'' - 1, s42 >= 0, s42 <= s12, s43 >= 0, s43 <= z'' - 1, s44 >= 0, s44 <= z'' - 1, s45 >= 0, s45 <= 0, s12 >= 0, s12 <= z'' - 1 + (z' - 1), s13 >= 0, s13 <= z'' - 1 + (z' - 1), s1 >= 0, s1 <= 0, z' - 1 >= 0, z >= 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 + s :|: s >= 0, s <= 0, z' - 1 >= 0, 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 + s20 :|: s20 >= 0, s20 <= 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 + s62 :|: s62 >= 0, s62 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 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 Function symbols to be analyzed: {MAX}, {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] -': runtime: O(n^1) [z'], size: O(1) [0] minus: runtime: O(1) [0], size: O(n^1) [z] min: runtime: O(1) [0], size: O(n^1) [z'] ---------------------------------------- (133) 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 ---------------------------------------- (134) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(s34, 1 + s35, z'') + s'' :|: s33 >= 0, s33 <= z' - 1, s34 >= 0, s34 <= s9, s35 >= 0, s35 <= z' - 1, s9 >= 0, s9 <= z' - 1 + (z - 1), s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(s60, z', 1 + s61) + s4 :|: s59 >= 0, s59 <= z'' - 1, s60 >= 0, s60 <= s19, s61 >= 0, s61 <= z'' - 1, s19 >= 0, s19 <= z'' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, s47, 1 + s48) + s2 :|: s46 >= 0, s46 <= z'' - 1, s47 >= 0, s47 <= s14, s48 >= 0, s48 <= z'' - 1, s14 >= 0, s14 <= z'' - 1 + (z' - 1), s2 >= 0, s2 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 + s26 }-> 1 + GCD(s24, 1 + s25, z'') + s27 + MAX(z - 1, z' - 1) :|: s23 >= 0, s23 <= z' - 1, s24 >= 0, s24 <= s5, s25 >= 0, s25 <= z' - 1, s26 >= 0, s26 <= z' - 1, s27 >= 0, s27 <= 0, s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ s31 + z' }-> 1 + GCD(s29, 1 + s30, z'') + s32 + s' :|: s28 >= 0, s28 <= z' - 1, s29 >= 0, s29 <= s7, s30 >= 0, s30 <= z' - 1, s31 >= 0, s31 <= z' - 1, s32 >= 0, s32 <= 0, s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 + s52 }-> 1 + GCD(s50, z', 1 + s51) + s53 + MAX(z - 1, z'' - 1) :|: s49 >= 0, s49 <= z'' - 1, s50 >= 0, s50 <= s15, s51 >= 0, s51 <= z'' - 1, s52 >= 0, s52 <= z'' - 1, s53 >= 0, s53 <= 0, s15 >= 0, s15 <= z'' - 1 + (z - 1), s16 >= 0, s16 <= z'' - 1 + (z - 1), z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ s57 + z'' }-> 1 + GCD(s55, z', 1 + s56) + s58 + s3 :|: s54 >= 0, s54 <= z'' - 1, s55 >= 0, s55 <= s17, s56 >= 0, s56 <= z'' - 1, s57 >= 0, s57 <= z'' - 1, s58 >= 0, s58 <= 0, s17 >= 0, s17 <= z'' - 1 + (z - 1), s18 >= 0, s18 <= z'' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 + s39 }-> 1 + GCD(z, s37, 1 + s38) + s40 + MAX(z' - 1, z'' - 1) :|: s36 >= 0, s36 <= z'' - 1, s37 >= 0, s37 <= s10, s38 >= 0, s38 <= z'' - 1, s39 >= 0, s39 <= z'' - 1, s40 >= 0, s40 <= 0, s10 >= 0, s10 <= z'' - 1 + (z' - 1), s11 >= 0, s11 <= z'' - 1 + (z' - 1), z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ s44 + z'' }-> 1 + GCD(z, s42, 1 + s43) + s45 + s1 :|: s41 >= 0, s41 <= z'' - 1, s42 >= 0, s42 <= s12, s43 >= 0, s43 <= z'' - 1, s44 >= 0, s44 <= z'' - 1, s45 >= 0, s45 <= 0, s12 >= 0, s12 <= z'' - 1 + (z' - 1), s13 >= 0, s13 <= z'' - 1 + (z' - 1), s1 >= 0, s1 <= 0, z' - 1 >= 0, z >= 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 + s :|: s >= 0, s <= 0, z' - 1 >= 0, 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 + s20 :|: s20 >= 0, s20 <= 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 + s62 :|: s62 >= 0, s62 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 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 Function symbols to be analyzed: {MAX}, {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] -': runtime: O(n^1) [z'], size: O(1) [0] minus: runtime: O(1) [0], size: O(n^1) [z] min: runtime: O(1) [0], size: O(n^1) [z'] MAX: runtime: ?, size: O(1) [0] ---------------------------------------- (135) 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' ---------------------------------------- (136) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(s34, 1 + s35, z'') + s'' :|: s33 >= 0, s33 <= z' - 1, s34 >= 0, s34 <= s9, s35 >= 0, s35 <= z' - 1, s9 >= 0, s9 <= z' - 1 + (z - 1), s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(s60, z', 1 + s61) + s4 :|: s59 >= 0, s59 <= z'' - 1, s60 >= 0, s60 <= s19, s61 >= 0, s61 <= z'' - 1, s19 >= 0, s19 <= z'' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, s47, 1 + s48) + s2 :|: s46 >= 0, s46 <= z'' - 1, s47 >= 0, s47 <= s14, s48 >= 0, s48 <= z'' - 1, s14 >= 0, s14 <= z'' - 1 + (z' - 1), s2 >= 0, s2 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 + s26 }-> 1 + GCD(s24, 1 + s25, z'') + s27 + MAX(z - 1, z' - 1) :|: s23 >= 0, s23 <= z' - 1, s24 >= 0, s24 <= s5, s25 >= 0, s25 <= z' - 1, s26 >= 0, s26 <= z' - 1, s27 >= 0, s27 <= 0, s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ s31 + z' }-> 1 + GCD(s29, 1 + s30, z'') + s32 + s' :|: s28 >= 0, s28 <= z' - 1, s29 >= 0, s29 <= s7, s30 >= 0, s30 <= z' - 1, s31 >= 0, s31 <= z' - 1, s32 >= 0, s32 <= 0, s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ 1 + s52 }-> 1 + GCD(s50, z', 1 + s51) + s53 + MAX(z - 1, z'' - 1) :|: s49 >= 0, s49 <= z'' - 1, s50 >= 0, s50 <= s15, s51 >= 0, s51 <= z'' - 1, s52 >= 0, s52 <= z'' - 1, s53 >= 0, s53 <= 0, s15 >= 0, s15 <= z'' - 1 + (z - 1), s16 >= 0, s16 <= z'' - 1 + (z - 1), z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ s57 + z'' }-> 1 + GCD(s55, z', 1 + s56) + s58 + s3 :|: s54 >= 0, s54 <= z'' - 1, s55 >= 0, s55 <= s17, s56 >= 0, s56 <= z'' - 1, s57 >= 0, s57 <= z'' - 1, s58 >= 0, s58 <= 0, s17 >= 0, s17 <= z'' - 1 + (z - 1), s18 >= 0, s18 <= z'' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ 1 + s39 }-> 1 + GCD(z, s37, 1 + s38) + s40 + MAX(z' - 1, z'' - 1) :|: s36 >= 0, s36 <= z'' - 1, s37 >= 0, s37 <= s10, s38 >= 0, s38 <= z'' - 1, s39 >= 0, s39 <= z'' - 1, s40 >= 0, s40 <= 0, s10 >= 0, s10 <= z'' - 1 + (z' - 1), s11 >= 0, s11 <= z'' - 1 + (z' - 1), z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ s44 + z'' }-> 1 + GCD(z, s42, 1 + s43) + s45 + s1 :|: s41 >= 0, s41 <= z'' - 1, s42 >= 0, s42 <= s12, s43 >= 0, s43 <= z'' - 1, s44 >= 0, s44 <= z'' - 1, s45 >= 0, s45 <= 0, s12 >= 0, s12 <= z'' - 1 + (z' - 1), s13 >= 0, s13 <= z'' - 1 + (z' - 1), s1 >= 0, s1 <= 0, z' - 1 >= 0, z >= 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 + s :|: s >= 0, s <= 0, z' - 1 >= 0, 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 + s20 :|: s20 >= 0, s20 <= 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 + s62 :|: s62 >= 0, s62 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 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 Function symbols to be analyzed: {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] -': runtime: O(n^1) [z'], size: O(1) [0] minus: runtime: O(1) [0], size: O(n^1) [z] min: runtime: O(1) [0], size: O(n^1) [z'] MAX: runtime: O(n^1) [z'], size: O(1) [0] ---------------------------------------- (137) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (138) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(s34, 1 + s35, z'') + s'' :|: s33 >= 0, s33 <= z' - 1, s34 >= 0, s34 <= s9, s35 >= 0, s35 <= z' - 1, s9 >= 0, s9 <= z' - 1 + (z - 1), s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(s60, z', 1 + s61) + s4 :|: s59 >= 0, s59 <= z'' - 1, s60 >= 0, s60 <= s19, s61 >= 0, s61 <= z'' - 1, s19 >= 0, s19 <= z'' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, s47, 1 + s48) + s2 :|: s46 >= 0, s46 <= z'' - 1, s47 >= 0, s47 <= s14, s48 >= 0, s48 <= z'' - 1, s14 >= 0, s14 <= z'' - 1 + (z' - 1), s2 >= 0, s2 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ s26 + z' }-> 1 + GCD(s24, 1 + s25, z'') + s27 + s64 :|: s64 >= 0, s64 <= 0, s23 >= 0, s23 <= z' - 1, s24 >= 0, s24 <= s5, s25 >= 0, s25 <= z' - 1, s26 >= 0, s26 <= z' - 1, s27 >= 0, s27 <= 0, s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ s31 + z' }-> 1 + GCD(s29, 1 + s30, z'') + s32 + s' :|: s28 >= 0, s28 <= z' - 1, s29 >= 0, s29 <= s7, s30 >= 0, s30 <= z' - 1, s31 >= 0, s31 <= z' - 1, s32 >= 0, s32 <= 0, s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ s52 + z'' }-> 1 + GCD(s50, z', 1 + s51) + s53 + s66 :|: s66 >= 0, s66 <= 0, s49 >= 0, s49 <= z'' - 1, s50 >= 0, s50 <= s15, s51 >= 0, s51 <= z'' - 1, s52 >= 0, s52 <= z'' - 1, s53 >= 0, s53 <= 0, s15 >= 0, s15 <= z'' - 1 + (z - 1), s16 >= 0, s16 <= z'' - 1 + (z - 1), z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ s57 + z'' }-> 1 + GCD(s55, z', 1 + s56) + s58 + s3 :|: s54 >= 0, s54 <= z'' - 1, s55 >= 0, s55 <= s17, s56 >= 0, s56 <= z'' - 1, s57 >= 0, s57 <= z'' - 1, s58 >= 0, s58 <= 0, s17 >= 0, s17 <= z'' - 1 + (z - 1), s18 >= 0, s18 <= z'' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ s39 + z'' }-> 1 + GCD(z, s37, 1 + s38) + s40 + s65 :|: s65 >= 0, s65 <= 0, s36 >= 0, s36 <= z'' - 1, s37 >= 0, s37 <= s10, s38 >= 0, s38 <= z'' - 1, s39 >= 0, s39 <= z'' - 1, s40 >= 0, s40 <= 0, s10 >= 0, s10 <= z'' - 1 + (z' - 1), s11 >= 0, s11 <= z'' - 1 + (z' - 1), z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ s44 + z'' }-> 1 + GCD(z, s42, 1 + s43) + s45 + s1 :|: s41 >= 0, s41 <= z'' - 1, s42 >= 0, s42 <= s12, s43 >= 0, s43 <= z'' - 1, s44 >= 0, s44 <= z'' - 1, s45 >= 0, s45 <= 0, s12 >= 0, s12 <= z'' - 1 + (z' - 1), s13 >= 0, s13 <= z'' - 1 + (z' - 1), s1 >= 0, s1 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 MAX(z, z') -{ z' }-> 1 + s63 :|: s63 >= 0, s63 <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, 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 + s20 :|: s20 >= 0, s20 <= 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 + s62 :|: s62 >= 0, s62 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 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 Function symbols to be analyzed: {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] -': runtime: O(n^1) [z'], size: O(1) [0] minus: runtime: O(1) [0], size: O(n^1) [z] min: runtime: O(1) [0], size: O(n^1) [z'] MAX: runtime: O(n^1) [z'], size: O(1) [0] ---------------------------------------- (139) 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 ---------------------------------------- (140) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(s34, 1 + s35, z'') + s'' :|: s33 >= 0, s33 <= z' - 1, s34 >= 0, s34 <= s9, s35 >= 0, s35 <= z' - 1, s9 >= 0, s9 <= z' - 1 + (z - 1), s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(s60, z', 1 + s61) + s4 :|: s59 >= 0, s59 <= z'' - 1, s60 >= 0, s60 <= s19, s61 >= 0, s61 <= z'' - 1, s19 >= 0, s19 <= z'' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, s47, 1 + s48) + s2 :|: s46 >= 0, s46 <= z'' - 1, s47 >= 0, s47 <= s14, s48 >= 0, s48 <= z'' - 1, s14 >= 0, s14 <= z'' - 1 + (z' - 1), s2 >= 0, s2 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ s26 + z' }-> 1 + GCD(s24, 1 + s25, z'') + s27 + s64 :|: s64 >= 0, s64 <= 0, s23 >= 0, s23 <= z' - 1, s24 >= 0, s24 <= s5, s25 >= 0, s25 <= z' - 1, s26 >= 0, s26 <= z' - 1, s27 >= 0, s27 <= 0, s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ s31 + z' }-> 1 + GCD(s29, 1 + s30, z'') + s32 + s' :|: s28 >= 0, s28 <= z' - 1, s29 >= 0, s29 <= s7, s30 >= 0, s30 <= z' - 1, s31 >= 0, s31 <= z' - 1, s32 >= 0, s32 <= 0, s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ s52 + z'' }-> 1 + GCD(s50, z', 1 + s51) + s53 + s66 :|: s66 >= 0, s66 <= 0, s49 >= 0, s49 <= z'' - 1, s50 >= 0, s50 <= s15, s51 >= 0, s51 <= z'' - 1, s52 >= 0, s52 <= z'' - 1, s53 >= 0, s53 <= 0, s15 >= 0, s15 <= z'' - 1 + (z - 1), s16 >= 0, s16 <= z'' - 1 + (z - 1), z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ s57 + z'' }-> 1 + GCD(s55, z', 1 + s56) + s58 + s3 :|: s54 >= 0, s54 <= z'' - 1, s55 >= 0, s55 <= s17, s56 >= 0, s56 <= z'' - 1, s57 >= 0, s57 <= z'' - 1, s58 >= 0, s58 <= 0, s17 >= 0, s17 <= z'' - 1 + (z - 1), s18 >= 0, s18 <= z'' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ s39 + z'' }-> 1 + GCD(z, s37, 1 + s38) + s40 + s65 :|: s65 >= 0, s65 <= 0, s36 >= 0, s36 <= z'' - 1, s37 >= 0, s37 <= s10, s38 >= 0, s38 <= z'' - 1, s39 >= 0, s39 <= z'' - 1, s40 >= 0, s40 <= 0, s10 >= 0, s10 <= z'' - 1 + (z' - 1), s11 >= 0, s11 <= z'' - 1 + (z' - 1), z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ s44 + z'' }-> 1 + GCD(z, s42, 1 + s43) + s45 + s1 :|: s41 >= 0, s41 <= z'' - 1, s42 >= 0, s42 <= s12, s43 >= 0, s43 <= z'' - 1, s44 >= 0, s44 <= z'' - 1, s45 >= 0, s45 <= 0, s12 >= 0, s12 <= z'' - 1 + (z' - 1), s13 >= 0, s13 <= z'' - 1 + (z' - 1), s1 >= 0, s1 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 MAX(z, z') -{ z' }-> 1 + s63 :|: s63 >= 0, s63 <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, 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 + s20 :|: s20 >= 0, s20 <= 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 + s62 :|: s62 >= 0, s62 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 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 Function symbols to be analyzed: {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] -': runtime: O(n^1) [z'], size: O(1) [0] minus: runtime: O(1) [0], size: O(n^1) [z] min: runtime: O(1) [0], size: O(n^1) [z'] MAX: runtime: O(n^1) [z'], size: O(1) [0] GCD: runtime: ?, size: O(1) [0] ---------------------------------------- (141) 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: ? ---------------------------------------- (142) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0 GCD(z, z', z'') -{ z' }-> 1 + GCD(s34, 1 + s35, z'') + s'' :|: s33 >= 0, s33 <= z' - 1, s34 >= 0, s34 <= s9, s35 >= 0, s35 <= z' - 1, s9 >= 0, s9 <= z' - 1 + (z - 1), s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(s60, z', 1 + s61) + s4 :|: s59 >= 0, s59 <= z'' - 1, s60 >= 0, s60 <= s19, s61 >= 0, s61 <= z'' - 1, s19 >= 0, s19 <= z'' - 1 + (z - 1), s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ z'' }-> 1 + GCD(z, s47, 1 + s48) + s2 :|: s46 >= 0, s46 <= z'' - 1, s47 >= 0, s47 <= s14, s48 >= 0, s48 <= z'' - 1, s14 >= 0, s14 <= z'' - 1 + (z' - 1), s2 >= 0, s2 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ s26 + z' }-> 1 + GCD(s24, 1 + s25, z'') + s27 + s64 :|: s64 >= 0, s64 <= 0, s23 >= 0, s23 <= z' - 1, s24 >= 0, s24 <= s5, s25 >= 0, s25 <= z' - 1, s26 >= 0, s26 <= z' - 1, s27 >= 0, s27 <= 0, s5 >= 0, s5 <= z' - 1 + (z - 1), s6 >= 0, s6 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ s31 + z' }-> 1 + GCD(s29, 1 + s30, z'') + s32 + s' :|: s28 >= 0, s28 <= z' - 1, s29 >= 0, s29 <= s7, s30 >= 0, s30 <= z' - 1, s31 >= 0, s31 <= z' - 1, s32 >= 0, s32 <= 0, s7 >= 0, s7 <= z' - 1 + (z - 1), s8 >= 0, s8 <= z' - 1 + (z - 1), s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 GCD(z, z', z'') -{ s52 + z'' }-> 1 + GCD(s50, z', 1 + s51) + s53 + s66 :|: s66 >= 0, s66 <= 0, s49 >= 0, s49 <= z'' - 1, s50 >= 0, s50 <= s15, s51 >= 0, s51 <= z'' - 1, s52 >= 0, s52 <= z'' - 1, s53 >= 0, s53 <= 0, s15 >= 0, s15 <= z'' - 1 + (z - 1), s16 >= 0, s16 <= z'' - 1 + (z - 1), z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ s57 + z'' }-> 1 + GCD(s55, z', 1 + s56) + s58 + s3 :|: s54 >= 0, s54 <= z'' - 1, s55 >= 0, s55 <= s17, s56 >= 0, s56 <= z'' - 1, s57 >= 0, s57 <= z'' - 1, s58 >= 0, s58 <= 0, s17 >= 0, s17 <= z'' - 1 + (z - 1), s18 >= 0, s18 <= z'' - 1 + (z - 1), s3 >= 0, s3 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ s39 + z'' }-> 1 + GCD(z, s37, 1 + s38) + s40 + s65 :|: s65 >= 0, s65 <= 0, s36 >= 0, s36 <= z'' - 1, s37 >= 0, s37 <= s10, s38 >= 0, s38 <= z'' - 1, s39 >= 0, s39 <= z'' - 1, s40 >= 0, s40 <= 0, s10 >= 0, s10 <= z'' - 1 + (z' - 1), s11 >= 0, s11 <= z'' - 1 + (z' - 1), z' - 1 >= 0, z >= 0, z'' - 1 >= 0 GCD(z, z', z'') -{ s44 + z'' }-> 1 + GCD(z, s42, 1 + s43) + s45 + s1 :|: s41 >= 0, s41 <= z'' - 1, s42 >= 0, s42 <= s12, s43 >= 0, s43 <= z'' - 1, s44 >= 0, s44 <= z'' - 1, s45 >= 0, s45 <= 0, s12 >= 0, s12 <= z'' - 1 + (z' - 1), s13 >= 0, s13 <= z'' - 1 + (z' - 1), s1 >= 0, s1 <= 0, z' - 1 >= 0, z >= 0, z'' - 1 >= 0 MAX(z, z') -{ z' }-> 1 + s63 :|: s63 >= 0, s63 <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, 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 + s20 :|: s20 >= 0, s20 <= 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 + s62 :|: s62 >= 0, s62 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 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 Function symbols to be analyzed: {GCD} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] max: runtime: O(1) [0], size: O(n^1) [z + z'] -': runtime: O(n^1) [z'], size: O(1) [0] minus: runtime: O(1) [0], size: O(n^1) [z] min: runtime: O(1) [0], size: O(n^1) [z'] MAX: runtime: O(n^1) [z'], size: O(1) [0] GCD: runtime: INF, size: O(1) [0] ---------------------------------------- (143) 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] max(v0, v1) -> null_max [0] min(v0, v1) -> null_min [0] MIN(v0, v1) -> null_MIN [0] MAX(v0, v1) -> null_MAX [0] -'(v0, v1) -> null_-' [0] GCD(v0, v1, v2) -> null_GCD [0] And the following fresh constants: null_minus, null_max, null_min, null_MIN, null_MAX, null_-', null_GCD ---------------------------------------- (144) 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] -'(s(z0), s(z1)) -> c7(-'(z0, z1)) [1] GCD(s(z0), s(z1), z2) -> c8(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) [1] GCD(s(z0), s(z1), z2) -> c9(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) [1] GCD(s(z0), s(z1), z2) -> c10(GCD(minus(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) [1] GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, minus(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) [1] GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, minus(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) [1] GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, minus(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) [1] GCD(s(z0), z1, s(z2)) -> c14(GCD(minus(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) [1] GCD(s(z0), z1, s(z2)) -> c15(GCD(minus(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) [1] GCD(s(z0), z1, s(z2)) -> c16(GCD(minus(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) [1] minus(z0, 0) -> z0 [0] minus(s(z0), s(z1)) -> minus(z0, 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] 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] -'(v0, v1) -> null_-' [0] GCD(v0, v1, v2) -> null_GCD [0] The TRS has the following type information: MIN :: s:0:null_minus:null_max:null_min -> s:0:null_minus:null_max:null_min -> c2:null_MIN s :: s:0:null_minus:null_max:null_min -> s:0:null_minus:null_max:null_min c2 :: c2:null_MIN -> c2:null_MIN MAX :: s:0:null_minus:null_max:null_min -> s:0:null_minus:null_max:null_min -> c5:null_MAX c5 :: c5:null_MAX -> c5:null_MAX -' :: s:0:null_minus:null_max:null_min -> s:0:null_minus:null_max:null_min -> c7:null_-' c7 :: c7:null_-' -> c7:null_-' GCD :: s:0:null_minus:null_max:null_min -> s:0:null_minus:null_max:null_min -> s:0:null_minus:null_max:null_min -> c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD c8 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD -> c7:null_-' -> c5:null_MAX -> c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD minus :: s:0:null_minus:null_max:null_min -> s:0:null_minus:null_max:null_min -> s:0:null_minus:null_max:null_min max :: s:0:null_minus:null_max:null_min -> s:0:null_minus:null_max:null_min -> s:0:null_minus:null_max:null_min min :: s:0:null_minus:null_max:null_min -> s:0:null_minus:null_max:null_min -> s:0:null_minus:null_max:null_min c9 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD -> c7:null_-' -> c2:null_MIN -> c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD c10 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD -> c2:null_MIN -> c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD c11 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD -> c7:null_-' -> c5:null_MAX -> c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD c12 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD -> c7:null_-' -> c2:null_MIN -> c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD c13 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD -> c2:null_MIN -> c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD c14 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD -> c7:null_-' -> c5:null_MAX -> c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD c15 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD -> c7:null_-' -> c2:null_MIN -> c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD c16 :: c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD -> c2:null_MIN -> c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD 0 :: s:0:null_minus:null_max:null_min null_minus :: s:0:null_minus:null_max:null_min null_max :: s:0:null_minus:null_max:null_min null_min :: s:0:null_minus:null_max:null_min null_MIN :: c2:null_MIN null_MAX :: c5:null_MAX null_-' :: c7:null_-' null_GCD :: c8:c9:c10:c11:c12:c13:c14:c15:c16:null_GCD Rewrite Strategy: INNERMOST ---------------------------------------- (145) 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_max => 0 null_min => 0 null_MIN => 0 null_MAX => 0 null_-' => 0 null_GCD => 0 ---------------------------------------- (146) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 -'(z, z') -{ 1 }-> 1 + -'(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 GCD(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z0, minus(max(z1, z2), min(z1, z2)), 1 + min(z1, z2)) + MIN(z1, z2) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1, z'' = 1 + z2, z2 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z0, z1), min(z0, z1)), 1 + min(z0, z1), z2) + MIN(z0, z1) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z2 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z0, z2), min(z0, z2)), z1, 1 + min(z0, z2)) + MIN(z0, z2) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0, z'' = 1 + z2, z2 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z0, minus(max(z1, z2), min(z1, z2)), 1 + min(z1, z2)) + -'(max(z1, z2), min(z1, z2)) + MIN(z1, z2) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1, z'' = 1 + z2, z2 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(z0, minus(max(z1, z2), min(z1, z2)), 1 + min(z1, z2)) + -'(max(z1, z2), min(z1, z2)) + MAX(z1, z2) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1, z'' = 1 + z2, z2 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z0, z1), min(z0, z1)), 1 + min(z0, z1), z2) + -'(max(z0, z1), min(z0, z1)) + MIN(z0, z1) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z2 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z0, z1), min(z0, z1)), 1 + min(z0, z1), z2) + -'(max(z0, z1), min(z0, z1)) + MAX(z0, z1) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z2 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z0, z2), min(z0, z2)), z1, 1 + min(z0, z2)) + -'(max(z0, z2), min(z0, z2)) + MIN(z0, z2) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0, z'' = 1 + z2, z2 >= 0 GCD(z, z', z'') -{ 1 }-> 1 + GCD(minus(max(z0, z2), min(z0, z2)), z1, 1 + min(z0, z2)) + -'(max(z0, z2), min(z0, z2)) + MAX(z0, z2) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0, z'' = 1 + z2, z2 >= 0 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 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 }-> minus(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 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (147) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1)) by GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(max(z0, 0), 0), MAX(z0, 0)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(max(0, z0), 0), MAX(0, z0)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0)), MAX(z0, 0)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0)), MAX(0, z0)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(max(z0, 0), 0), MAX(z0, 0)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(max(0, z0), 0), MAX(0, z0)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0)), MAX(z0, 0)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0)), MAX(0, z0)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(max(z0, 0), 0), MAX(z0, 0)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(max(0, z0), 0), MAX(0, z0)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0)), MAX(z0, 0)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0)), MAX(0, z0)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c9_3, c10_2, c11_3, c12_3, c13_2, c14_3, c15_3, c16_2, c8_3, c8_1 ---------------------------------------- (149) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c9_3, c10_2, c11_3, c12_3, c13_2, c14_3, c15_3, c16_2, c8_3, c8_1, c8_2 ---------------------------------------- (151) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1)) by GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(max(z0, 0), 0), MIN(z0, 0)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(max(0, z0), 0), MIN(0, z0)) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0)), MIN(z0, 0)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0)), MIN(0, z0)) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(max(z0, 0), 0), MIN(z0, 0)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(max(0, z0), 0), MIN(0, z0)) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0)), MIN(z0, 0)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0)), MIN(0, z0)) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(max(z0, 0), 0), MIN(z0, 0)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(max(0, z0), 0), MIN(0, z0)) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0)), MIN(z0, 0)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0)), MIN(0, z0)) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c11_3, c12_3, c13_2, c14_3, c15_3, c16_2, c8_3, c8_1, c8_2, c9_3, c9_1 ---------------------------------------- (153) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c11_3, c12_3, c13_2, c14_3, c15_3, c16_2, c8_3, c8_1, c8_2, c9_3, c9_1, c9_2 ---------------------------------------- (155) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2)) by GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(max(z0, 0), 0), MAX(z0, 0)) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(max(0, z0), 0), MAX(0, z0)) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0)), MAX(z0, 0)) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0)), MAX(0, z0)) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(max(z0, 0), 0), MAX(z0, 0)) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(max(0, z0), 0), MAX(0, z0)) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0)), MAX(z0, 0)) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0)), MAX(0, z0)) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(max(z0, 0), 0), MAX(z0, 0)) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(max(0, z0), 0), MAX(0, z0)) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0)), MAX(z0, 0)) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0)), MAX(0, z0)) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c12_3, c13_2, c14_3, c15_3, c16_2, c8_3, c8_1, c8_2, c9_3, c9_1, c9_2, c11_3, c11_1 ---------------------------------------- (157) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c12_3, c13_2, c14_3, c15_3, c16_2, c8_3, c8_1, c8_2, c9_3, c9_1, c9_2, c11_3, c11_1, c11_2 ---------------------------------------- (159) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2)) by GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(max(z0, 0), 0), MIN(z0, 0)) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(max(0, z0), 0), MIN(0, z0)) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0)), MIN(z0, 0)) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0)), MIN(0, z0)) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(max(z0, 0), 0), MIN(z0, 0)) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(max(0, z0), 0), MIN(0, z0)) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0)), MIN(z0, 0)) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0)), MIN(0, z0)) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(max(z0, 0), 0), MIN(z0, 0)) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(max(0, z0), 0), MIN(0, z0)) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0)), MIN(z0, 0)) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0)), MIN(0, z0)) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c14_3, c15_3, c16_2, c8_3, c8_1, c8_2, c9_3, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1 ---------------------------------------- (161) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c14_3, c15_3, c16_2, c8_3, c8_1, c8_2, c9_3, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2 ---------------------------------------- (163) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2)) by GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(max(z0, 0), 0), MAX(z0, 0)) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(max(0, z0), 0), MAX(0, z0)) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0)), MAX(z0, 0)) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0)), MAX(0, z0)) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(max(z0, 0), 0), MAX(z0, 0)) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(max(0, z0), 0), MAX(0, z0)) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0)), MAX(z0, 0)) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0)), MAX(0, z0)) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(max(z0, 0), 0), MAX(z0, 0)) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(max(0, z0), 0), MAX(0, z0)) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0)), MAX(z0, 0)) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0)), MAX(0, z0)) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c15_3, c16_2, c8_3, c8_1, c8_2, c9_3, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1 ---------------------------------------- (165) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c15_3, c16_2, c8_3, c8_1, c8_2, c9_3, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2 ---------------------------------------- (167) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2)) by GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(max(z0, 0), 0), MIN(z0, 0)) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(max(0, z0), 0), MIN(0, z0)) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0)), MIN(z0, 0)) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0)), MIN(0, z0)) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(max(z0, 0), 0), MIN(z0, 0)) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(max(0, z0), 0), MIN(0, z0)) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0)), MIN(z0, 0)) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0)), MIN(0, z0)) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(max(z0, 0), 0), MIN(z0, 0)) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(max(0, z0), 0), MIN(0, z0)) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0)), MIN(z0, 0)) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0)), MIN(0, z0)) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_3, c8_1, c8_2, c9_3, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1 ---------------------------------------- (169) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_3, c8_1, c8_2, c9_3, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2 ---------------------------------------- (171) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) by GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_3, c8_1, c8_2, c9_3, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2 ---------------------------------------- (173) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) by GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c8_2, c9_3, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3 ---------------------------------------- (175) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) by GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c8_2, c9_3, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3 ---------------------------------------- (177) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) by GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c8_2, c9_3, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3 ---------------------------------------- (179) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), s(0), x2) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) by GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1), -'(0, min(0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1), -'(0, min(0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1), -'(0, min(0, 0))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1), -'(0, min(0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1), -'(0, min(0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1), -'(0, min(0, 0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1), -'(0, min(0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1), -'(0, min(0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1), -'(0, min(0, 0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c8_2, c9_3, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3 ---------------------------------------- (181) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c8_2, c9_3, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3 ---------------------------------------- (183) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(0), s(z0), x2) -> c8(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) by GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1), -'(0, min(0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1), -'(0, min(0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1), -'(0, min(0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1), -'(0, min(0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1), -'(0, min(0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1), -'(0, min(0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1), -'(0, min(0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1), -'(0, min(0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1), -'(0, min(0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_3, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2 ---------------------------------------- (185) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_3, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2 ---------------------------------------- (187) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) by GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_3, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2 ---------------------------------------- (189) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3 ---------------------------------------- (191) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2)) by GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3 ---------------------------------------- (193) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2)) by GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3 ---------------------------------------- (195) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), s(0), x2) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(min(z0, 0)), x2), -'(z0, min(z0, 0))) by GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1), -'(0, min(0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1), -'(0, min(0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1), -'(0, min(0, 0))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1), -'(0, min(0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1), -'(0, min(0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1), -'(0, min(0, 0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1), -'(0, min(0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1), -'(0, min(0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1), -'(0, min(0, 0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3 ---------------------------------------- (197) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c9_2, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3 ---------------------------------------- (199) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(0), s(z0), x2) -> c9(GCD(-(max(0, z0), min(0, z0)), s(min(0, z0)), x2), -'(z0, min(0, z0))) by GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1), -'(0, min(0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1), -'(0, min(0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1), -'(0, min(0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1), -'(0, min(0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1), -'(0, min(0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1), -'(0, min(0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1), -'(0, min(0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1), -'(0, min(0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1), -'(0, min(0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2 ---------------------------------------- (201) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2 ---------------------------------------- (203) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) by GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_3, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2 ---------------------------------------- (205) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) by GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3 ---------------------------------------- (207) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) by GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3 ---------------------------------------- (209) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) by GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3 ---------------------------------------- (211) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) by GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0)), -'(0, min(0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0))), -'(0, min(0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0))), -'(0, min(0, 0))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0)), -'(0, min(0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0))), -'(0, min(0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0)), -'(0, min(0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0))), -'(0, min(0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0))), -'(0, min(0, 0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3 ---------------------------------------- (213) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c11_2, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3 ---------------------------------------- (215) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) by GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0)), -'(0, min(0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0))), -'(0, min(0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0))), -'(0, min(0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0)), -'(0, min(0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0))), -'(0, min(0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0))), -'(0, min(0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0)), -'(0, min(0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0))), -'(0, min(0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0))), -'(0, min(0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2 ---------------------------------------- (217) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2 ---------------------------------------- (219) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) by GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_3, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2 ---------------------------------------- (221) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3 ---------------------------------------- (223) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0)))) by GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) ---------------------------------------- (224) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3 ---------------------------------------- (225) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0)))) by GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) ---------------------------------------- (226) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3 ---------------------------------------- (227) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) by GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0)), -'(0, min(0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0))), -'(0, min(0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0))), -'(0, min(0, 0))) ---------------------------------------- (228) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0)), -'(0, min(0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0))), -'(0, min(0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0)), -'(0, min(0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0))), -'(0, min(0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0))), -'(0, min(0, 0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3 ---------------------------------------- (229) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (230) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c12_2, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3 ---------------------------------------- (231) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) by GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0)), -'(0, min(0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0))), -'(0, min(0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0))), -'(0, min(0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) ---------------------------------------- (232) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0)), -'(0, min(0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0))), -'(0, min(0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0))), -'(0, min(0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0)), -'(0, min(0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0))), -'(0, min(0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0))), -'(0, min(0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2 ---------------------------------------- (233) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (234) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2 ---------------------------------------- (235) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) by GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) ---------------------------------------- (236) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_3, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2 ---------------------------------------- (237) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) by GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) ---------------------------------------- (238) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2, c14_3 ---------------------------------------- (239) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) by GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) ---------------------------------------- (240) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2, c14_3 ---------------------------------------- (241) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) by GCD(s(0), x0, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(0)) -> c14(GCD(-(max(0, 0), 0), x0, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(0)) -> c14(GCD(-(0, min(0, 0)), x0, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) ---------------------------------------- (242) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2, c14_3 ---------------------------------------- (243) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) by GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0)), -'(0, min(0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0))), -'(0, min(0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0))), -'(0, min(0, 0))) ---------------------------------------- (244) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0)), -'(0, min(0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0))), -'(0, min(0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0)), -'(0, min(0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0))), -'(0, min(0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0))), -'(0, min(0, 0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2, c14_3 ---------------------------------------- (245) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (246) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_1, c14_2, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2, c14_3 ---------------------------------------- (247) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(0), x1, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) by GCD(s(0), x0, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x0, s(0)), -'(0, min(0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(0)) -> c14(GCD(-(max(0, 0), 0), x0, s(min(0, 0))), -'(0, min(0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(0)) -> c14(GCD(-(0, min(0, 0)), x0, s(min(0, 0))), -'(0, min(0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) ---------------------------------------- (248) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x0, s(0)), -'(0, min(0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(0)) -> c14(GCD(-(max(0, 0), 0), x0, s(min(0, 0))), -'(0, min(0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(0)) -> c14(GCD(-(0, min(0, 0)), x0, s(min(0, 0))), -'(0, min(0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x0, s(0)), -'(0, min(0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(0)) -> c14(GCD(-(max(0, 0), 0), x0, s(min(0, 0))), -'(0, min(0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(0)) -> c14(GCD(-(0, min(0, 0)), x0, s(min(0, 0))), -'(0, min(0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_1, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2, c14_3, c14_2 ---------------------------------------- (249) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (250) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_1, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2, c14_3, c14_2 ---------------------------------------- (251) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) by GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) ---------------------------------------- (252) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_1, c15_3, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2, c14_3, c14_2 ---------------------------------------- (253) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (254) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_1, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2, c14_3, c14_2, c15_3 ---------------------------------------- (255) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0)))) by GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) ---------------------------------------- (256) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_1, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2, c14_3, c14_2, c15_3 ---------------------------------------- (257) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0)))) by GCD(s(0), x0, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(0)) -> c15(GCD(-(max(0, 0), 0), x0, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(0)) -> c15(GCD(-(0, min(0, 0)), x0, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) ---------------------------------------- (258) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_1, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2, c14_3, c14_2, c15_3 ---------------------------------------- (259) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) by GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x1, s(0)), -'(0, min(0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), 0), x1, s(min(0, 0))), -'(0, min(0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(0, min(0, 0)), x1, s(min(0, 0))), -'(0, min(0, 0))) ---------------------------------------- (260) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x1, s(0)), -'(0, min(0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), 0), x1, s(min(0, 0))), -'(0, min(0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(0, min(0, 0)), x1, s(min(0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x1, s(0)), -'(0, min(0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), 0), x1, s(min(0, 0))), -'(0, min(0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(0, min(0, 0)), x1, s(min(0, 0))), -'(0, min(0, 0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_1, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2, c14_3, c14_2, c15_3 ---------------------------------------- (261) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (262) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_1, c15_1, c15_2, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2, c14_3, c14_2, c15_3 ---------------------------------------- (263) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(s(0), x1, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x1, s(min(0, z0))), -'(z0, min(0, z0))) by GCD(s(0), x0, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x0, s(0)), -'(0, min(0, 0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(0)) -> c15(GCD(-(max(0, 0), 0), x0, s(min(0, 0))), -'(0, min(0, 0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(0)) -> c15(GCD(-(0, min(0, 0)), x0, s(min(0, 0))), -'(0, min(0, 0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) ---------------------------------------- (264) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x0, s(0)), -'(0, min(0, 0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(0)) -> c15(GCD(-(max(0, 0), 0), x0, s(min(0, 0))), -'(0, min(0, 0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(0)) -> c15(GCD(-(0, min(0, 0)), x0, s(min(0, 0))), -'(0, min(0, 0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x0, s(0)), -'(0, min(0, 0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(0)) -> c15(GCD(-(max(0, 0), 0), x0, s(min(0, 0))), -'(0, min(0, 0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(0)) -> c15(GCD(-(0, min(0, 0)), x0, s(min(0, 0))), -'(0, min(0, 0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_1, c15_1, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2, c14_3, c14_2, c15_3, c15_2 ---------------------------------------- (265) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (266) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MIN_2, MAX_2, -'_2, GCD_3 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_1, c15_1, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2, c14_3, c14_2, c15_3, c15_2 ---------------------------------------- (267) 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))) ---------------------------------------- (268) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, 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)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1)) GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2)) GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2)) GCD(s(x0), s(x1), x2) -> c8(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(s(x0), s(x1), x2) -> c9(GCD(-(max(x0, x1), min(x0, x1)), s(min(x0, x1)), x2)) GCD(x0, s(x1), s(x2)) -> c11(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(x0, s(x1), s(x2)) -> c12(GCD(x0, -(max(x1, x2), min(x1, x2)), s(min(x1, x2)))) GCD(s(x0), x1, s(x2)) -> c14(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(x0), x1, s(x2)) -> c15(GCD(-(max(x0, x2), min(x0, x2)), x1, s(min(x0, x2)))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c8(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c8(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c8(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c8(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), s(s(z1)), x2) -> c9(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1))), x2), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), min(0, 0)), s(0), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(max(0, 0), 0), s(min(0, 0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1)) GCD(s(0), s(0), x1) -> c9(GCD(-(0, min(0, 0)), s(min(0, 0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1)) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1)) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), min(z0, 0)), s(0), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(max(z0, 0), 0), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(z0), s(0), x1) -> c9(GCD(-(z0, min(z0, 0)), s(min(z0, 0)), x1), -'(z0, min(z0, 0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), min(0, z0)), s(0), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(max(0, z0), 0), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(s(0), s(z0), x1) -> c9(GCD(-(z0, min(0, z0)), s(min(0, z0)), x1), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c11(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c11(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c11(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c11(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), min(s(z0), s(z1))), s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(max(s(z0), s(z1)), s(min(z0, z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(s(z0)), s(s(z1))) -> c12(GCD(x0, -(s(max(z0, z1)), min(s(z0), s(z1))), s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), min(0, 0)), s(0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(max(0, 0), 0), s(min(0, 0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0)))) GCD(x0, s(0), s(0)) -> c12(GCD(x0, -(0, min(0, 0)), s(min(0, 0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0)))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0)))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), min(z0, 0)), s(0)), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(max(z0, 0), 0), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(z0), s(0)) -> c12(GCD(x0, -(z0, min(z0, 0)), s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), min(0, z0)), s(0)), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(max(0, z0), 0), s(min(0, z0))), -'(z0, min(0, z0))) GCD(x0, s(0), s(z0)) -> c12(GCD(x0, -(z0, min(0, z0)), s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c14(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c14(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c14(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c14(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(max(s(z0), s(z1)), s(min(z0, z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), min(s(z0), s(z1))), x1, s(s(min(z0, z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(max(s(z0), s(z1)), s(min(z0, z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(s(z0)), x1, s(s(z1))) -> c15(GCD(-(s(max(z0, z1)), min(s(z0), s(z1))), x1, s(min(s(z0), s(z1)))), -'(s(max(z0, z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), min(0, 0)), x1, s(0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(max(0, 0), 0), x1, s(min(0, 0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0)))) GCD(s(0), x1, s(0)) -> c15(GCD(-(0, min(0, 0)), x1, s(min(0, 0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0)))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0)))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), min(z0, 0)), x1, s(0)), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(max(z0, 0), 0), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(z0), x1, s(0)) -> c15(GCD(-(z0, min(z0, 0)), x1, s(min(z0, 0))), -'(z0, min(z0, 0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), min(0, z0)), x0, s(0)), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(max(0, z0), 0), x0, s(min(0, z0))), -'(z0, min(0, z0))) GCD(s(0), x0, s(z0)) -> c15(GCD(-(z0, min(0, z0)), x0, s(min(0, z0))), -'(z0, min(0, z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2 Defined Pair Symbols: MAX_2, -'_2, GCD_3, MIN_2 Compound Symbols: c5_1, c7_1, c10_2, c13_2, c16_2, c8_1, c9_1, c11_1, c12_1, c14_1, c15_1, c8_3, c8_2, c9_3, c9_2, c11_3, c11_2, c12_3, c12_2, c14_3, c14_2, c15_3, c15_2, c2_1