KILLED proof of input_B10DwyJIre.trs # AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 332 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), 41 ms] (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 57 ms] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 70 ms] (26) typed CpxTrs (27) RewriteLemmaProof [LOWER BOUND(ID), 72 ms] (28) typed CpxTrs (29) RewriteLemmaProof [LOWER BOUND(ID), 87 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), 1 ms] (44) CpxRNTS (45) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CpxRNTS (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 219 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 78 ms] (52) CpxRNTS (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 286 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 77 ms] (58) CpxRNTS (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 144 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 157 ms] (64) CpxRNTS (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (66) CpxRNTS (67) IntTrsBoundProof [UPPER BOUND(ID), 1876 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 1072 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), 18 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), 139 ms] (104) CpxRNTS (105) IntTrsBoundProof [UPPER BOUND(ID), 95 ms] (106) CpxRNTS (107) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (108) CpxRNTS (109) IntTrsBoundProof [UPPER BOUND(ID), 238 ms] (110) CpxRNTS (111) IntTrsBoundProof [UPPER BOUND(ID), 24 ms] (112) CpxRNTS (113) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (114) CpxRNTS (115) IntTrsBoundProof [UPPER BOUND(ID), 210 ms] (116) CpxRNTS (117) IntTrsBoundProof [UPPER BOUND(ID), 104 ms] (118) CpxRNTS (119) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (120) CpxRNTS (121) IntTrsBoundProof [UPPER BOUND(ID), 243 ms] (122) CpxRNTS (123) IntTrsBoundProof [UPPER BOUND(ID), 75 ms] (124) CpxRNTS (125) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (126) CpxRNTS (127) IntTrsBoundProof [UPPER BOUND(ID), 251 ms] (128) CpxRNTS (129) IntTrsBoundProof [UPPER BOUND(ID), 65 ms] (130) CpxRNTS (131) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (132) CpxRNTS (133) IntTrsBoundProof [UPPER BOUND(ID), 57 ms] (134) CpxRNTS (135) IntTrsBoundProof [UPPER BOUND(ID), 62 ms] (136) CpxRNTS (137) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (138) CpxRNTS (139) IntTrsBoundProof [UPPER BOUND(ID), 1932 ms] (140) CpxRNTS (141) IntTrsBoundProof [UPPER BOUND(ID), 872 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), 0 ms] (148) CdtProblem (149) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 120 ms] (194) CdtProblem (195) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 6 ms] (196) CdtProblem (197) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 15 ms] (198) CdtProblem (199) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms] (200) CdtProblem (201) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (210) CdtProblem (211) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (212) CdtProblem (213) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (216) CdtProblem (217) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 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), 0 ms] (226) CdtProblem (227) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 5 ms] (228) CdtProblem (229) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 194 ms] (230) CdtProblem (231) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (232) CdtProblem (233) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (234) CdtProblem (235) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (236) CdtProblem (237) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (238) CdtProblem (239) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (240) CdtProblem (241) CdtRewritingProof [BOTH BOUNDS(ID, ID), 5 ms] (242) CdtProblem (243) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1221 ms] (244) CdtProblem (245) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (246) CdtProblem (247) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 166 ms] (248) CdtProblem (249) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (250) CdtProblem (251) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (252) CdtProblem (253) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1196 ms] (254) CdtProblem (255) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (256) CdtProblem (257) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 166 ms] (258) CdtProblem (259) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (260) CdtProblem (261) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (262) CdtProblem (263) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (264) CdtProblem (265) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1133 ms] (266) CdtProblem (267) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (268) CdtProblem (269) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (270) CdtProblem (271) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1083 ms] (272) CdtProblem (273) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (274) CdtProblem (275) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (276) 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: -(x, 0) -> x -(s(x), s(y)) -> -(x, y) min(x, 0) -> 0 min(0, y) -> 0 min(s(x), s(y)) -> s(min(x, y)) twice(0) -> 0 twice(s(x)) -> s(s(twice(x))) f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y)))) f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y)))) 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: -(x, 0') -> x -(s(x), s(y)) -> -(x, y) min(x, 0') -> 0' min(0', y) -> 0' min(s(x), s(y)) -> s(min(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y)))) f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y)))) 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: -(x, 0) -> x -(s(x), s(y)) -> -(x, y) min(x, 0) -> 0 min(0, y) -> 0 min(s(x), s(y)) -> s(min(x, y)) twice(0) -> 0 twice(s(x)) -> s(s(twice(x))) f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y)))) f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y)))) 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: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1)))) f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1)))) Tuples: -'(z0, 0) -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(z0, 0) -> c2 MIN(0, z0) -> c3 MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(0) -> c5 TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) S tuples: -'(z0, 0) -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(z0, 0) -> c2 MIN(0, z0) -> c3 MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(0) -> c5 TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1, f_2 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c, c1_1, c2, c3, c4_1, c5, c6_1, c7_3, c8_3, c9_3, c10_3 ---------------------------------------- (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: -'(z0, 0) -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(z0, 0) -> c2 MIN(0, z0) -> c3 MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(0) -> c5 TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) The (relative) TRS S consists of the following rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1)))) f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1)))) 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: -'(z0, 0') -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(z0, 0') -> c2 MIN(0', z0) -> c3 MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(0') -> c5 TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) The (relative) TRS S consists of the following rules: -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0') -> 0' twice(s(z0)) -> s(s(twice(z0))) f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1)))) f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1)))) Rewrite Strategy: INNERMOST ---------------------------------------- (11) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (12) Obligation: Innermost TRS: Rules: -'(z0, 0') -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(z0, 0') -> c2 MIN(0', z0) -> c3 MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(0') -> c5 TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0') -> 0' twice(s(z0)) -> s(s(twice(z0))) f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1)))) f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1)))) Types: -' :: 0':s -> 0':s -> c:c1 0' :: 0':s c :: c:c1 s :: 0':s -> 0':s c1 :: c:c1 -> c:c1 MIN :: 0':s -> 0':s -> c2:c3:c4 c2 :: c2:c3:c4 c3 :: c2:c3:c4 c4 :: c2:c3:c4 -> c2:c3:c4 TWICE :: 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 F :: 0':s -> 0':s -> c7:c8:c9:c10 c7 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10 - :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s twice :: 0':s -> 0':s c8 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10 c9 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10 c10 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10 f :: 0':s -> 0':s -> f hole_c:c11_11 :: c:c1 hole_0':s2_11 :: 0':s hole_c2:c3:c43_11 :: c2:c3:c4 hole_c5:c64_11 :: c5:c6 hole_c7:c8:c9:c105_11 :: c7:c8:c9:c10 hole_f6_11 :: f gen_c:c17_11 :: Nat -> c:c1 gen_0':s8_11 :: Nat -> 0':s gen_c2:c3:c49_11 :: Nat -> c2:c3:c4 gen_c5:c610_11 :: Nat -> c5:c6 gen_c7:c8:c9:c1011_11 :: Nat -> c7:c8:c9:c10 ---------------------------------------- (13) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: -', MIN, TWICE, F, -, min, twice, f They will be analysed ascendingly in the following order: -' < F MIN < F TWICE < F - < F min < F twice < F - < f min < f twice < f ---------------------------------------- (14) Obligation: Innermost TRS: Rules: -'(z0, 0') -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(z0, 0') -> c2 MIN(0', z0) -> c3 MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(0') -> c5 TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0') -> 0' twice(s(z0)) -> s(s(twice(z0))) f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1)))) f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1)))) Types: -' :: 0':s -> 0':s -> c:c1 0' :: 0':s c :: c:c1 s :: 0':s -> 0':s c1 :: c:c1 -> c:c1 MIN :: 0':s -> 0':s -> c2:c3:c4 c2 :: c2:c3:c4 c3 :: c2:c3:c4 c4 :: c2:c3:c4 -> c2:c3:c4 TWICE :: 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 F :: 0':s -> 0':s -> c7:c8:c9:c10 c7 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10 - :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s twice :: 0':s -> 0':s c8 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10 c9 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10 c10 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10 f :: 0':s -> 0':s -> f hole_c:c11_11 :: c:c1 hole_0':s2_11 :: 0':s hole_c2:c3:c43_11 :: c2:c3:c4 hole_c5:c64_11 :: c5:c6 hole_c7:c8:c9:c105_11 :: c7:c8:c9:c10 hole_f6_11 :: f gen_c:c17_11 :: Nat -> c:c1 gen_0':s8_11 :: Nat -> 0':s gen_c2:c3:c49_11 :: Nat -> c2:c3:c4 gen_c5:c610_11 :: Nat -> c5:c6 gen_c7:c8:c9:c1011_11 :: Nat -> c7:c8:c9:c10 Generator Equations: gen_c:c17_11(0) <=> c gen_c:c17_11(+(x, 1)) <=> c1(gen_c:c17_11(x)) gen_0':s8_11(0) <=> 0' gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x)) gen_c2:c3:c49_11(0) <=> c2 gen_c2:c3:c49_11(+(x, 1)) <=> c4(gen_c2:c3:c49_11(x)) gen_c5:c610_11(0) <=> c5 gen_c5:c610_11(+(x, 1)) <=> c6(gen_c5:c610_11(x)) gen_c7:c8:c9:c1011_11(0) <=> hole_c7:c8:c9:c105_11 gen_c7:c8:c9:c1011_11(+(x, 1)) <=> c7(gen_c7:c8:c9:c1011_11(x), c, c2) The following defined symbols remain to be analysed: -', MIN, TWICE, F, -, min, twice, f They will be analysed ascendingly in the following order: -' < F MIN < F TWICE < F - < F min < F twice < F - < f min < f twice < f ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: -'(gen_0':s8_11(n13_11), gen_0':s8_11(n13_11)) -> gen_c:c17_11(n13_11), rt in Omega(1 + n13_11) Induction Base: -'(gen_0':s8_11(0), gen_0':s8_11(0)) ->_R^Omega(1) c Induction Step: -'(gen_0':s8_11(+(n13_11, 1)), gen_0':s8_11(+(n13_11, 1))) ->_R^Omega(1) c1(-'(gen_0':s8_11(n13_11), gen_0':s8_11(n13_11))) ->_IH c1(gen_c:c17_11(c14_11)) 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: -'(z0, 0') -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(z0, 0') -> c2 MIN(0', z0) -> c3 MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(0') -> c5 TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0') -> 0' twice(s(z0)) -> s(s(twice(z0))) f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1)))) f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1)))) Types: -' :: 0':s -> 0':s -> c:c1 0' :: 0':s c :: c:c1 s :: 0':s -> 0':s c1 :: c:c1 -> c:c1 MIN :: 0':s -> 0':s -> c2:c3:c4 c2 :: c2:c3:c4 c3 :: c2:c3:c4 c4 :: c2:c3:c4 -> c2:c3:c4 TWICE :: 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 F :: 0':s -> 0':s -> c7:c8:c9:c10 c7 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10 - :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s twice :: 0':s -> 0':s c8 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10 c9 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10 c10 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10 f :: 0':s -> 0':s -> f hole_c:c11_11 :: c:c1 hole_0':s2_11 :: 0':s hole_c2:c3:c43_11 :: c2:c3:c4 hole_c5:c64_11 :: c5:c6 hole_c7:c8:c9:c105_11 :: c7:c8:c9:c10 hole_f6_11 :: f gen_c:c17_11 :: Nat -> c:c1 gen_0':s8_11 :: Nat -> 0':s gen_c2:c3:c49_11 :: Nat -> c2:c3:c4 gen_c5:c610_11 :: Nat -> c5:c6 gen_c7:c8:c9:c1011_11 :: Nat -> c7:c8:c9:c10 Generator Equations: gen_c:c17_11(0) <=> c gen_c:c17_11(+(x, 1)) <=> c1(gen_c:c17_11(x)) gen_0':s8_11(0) <=> 0' gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x)) gen_c2:c3:c49_11(0) <=> c2 gen_c2:c3:c49_11(+(x, 1)) <=> c4(gen_c2:c3:c49_11(x)) gen_c5:c610_11(0) <=> c5 gen_c5:c610_11(+(x, 1)) <=> c6(gen_c5:c610_11(x)) gen_c7:c8:c9:c1011_11(0) <=> hole_c7:c8:c9:c105_11 gen_c7:c8:c9:c1011_11(+(x, 1)) <=> c7(gen_c7:c8:c9:c1011_11(x), c, c2) The following defined symbols remain to be analysed: -', MIN, TWICE, F, -, min, twice, f They will be analysed ascendingly in the following order: -' < F MIN < F TWICE < F - < F min < F twice < F - < f min < f twice < f ---------------------------------------- (18) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (19) BOUNDS(n^1, INF) ---------------------------------------- (20) Obligation: Innermost TRS: Rules: -'(z0, 0') -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(z0, 0') -> c2 MIN(0', z0) -> c3 MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(0') -> c5 TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0') -> 0' twice(s(z0)) -> s(s(twice(z0))) f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1)))) f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1)))) Types: -' :: 0':s -> 0':s -> c:c1 0' :: 0':s c :: c:c1 s :: 0':s -> 0':s c1 :: c:c1 -> c:c1 MIN :: 0':s -> 0':s -> c2:c3:c4 c2 :: c2:c3:c4 c3 :: c2:c3:c4 c4 :: c2:c3:c4 -> c2:c3:c4 TWICE :: 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 F :: 0':s -> 0':s -> c7:c8:c9:c10 c7 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10 - :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s twice :: 0':s -> 0':s c8 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10 c9 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10 c10 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10 f :: 0':s -> 0':s -> f hole_c:c11_11 :: c:c1 hole_0':s2_11 :: 0':s hole_c2:c3:c43_11 :: c2:c3:c4 hole_c5:c64_11 :: c5:c6 hole_c7:c8:c9:c105_11 :: c7:c8:c9:c10 hole_f6_11 :: f gen_c:c17_11 :: Nat -> c:c1 gen_0':s8_11 :: Nat -> 0':s gen_c2:c3:c49_11 :: Nat -> c2:c3:c4 gen_c5:c610_11 :: Nat -> c5:c6 gen_c7:c8:c9:c1011_11 :: Nat -> c7:c8:c9:c10 Lemmas: -'(gen_0':s8_11(n13_11), gen_0':s8_11(n13_11)) -> gen_c:c17_11(n13_11), rt in Omega(1 + n13_11) Generator Equations: gen_c:c17_11(0) <=> c gen_c:c17_11(+(x, 1)) <=> c1(gen_c:c17_11(x)) gen_0':s8_11(0) <=> 0' gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x)) gen_c2:c3:c49_11(0) <=> c2 gen_c2:c3:c49_11(+(x, 1)) <=> c4(gen_c2:c3:c49_11(x)) gen_c5:c610_11(0) <=> c5 gen_c5:c610_11(+(x, 1)) <=> c6(gen_c5:c610_11(x)) gen_c7:c8:c9:c1011_11(0) <=> hole_c7:c8:c9:c105_11 gen_c7:c8:c9:c1011_11(+(x, 1)) <=> c7(gen_c7:c8:c9:c1011_11(x), c, c2) The following defined symbols remain to be analysed: MIN, TWICE, F, -, min, twice, f They will be analysed ascendingly in the following order: MIN < F TWICE < F - < F min < F twice < F - < f min < f twice < f ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: MIN(gen_0':s8_11(n426_11), gen_0':s8_11(n426_11)) -> gen_c2:c3:c49_11(n426_11), rt in Omega(1 + n426_11) Induction Base: MIN(gen_0':s8_11(0), gen_0':s8_11(0)) ->_R^Omega(1) c2 Induction Step: MIN(gen_0':s8_11(+(n426_11, 1)), gen_0':s8_11(+(n426_11, 1))) ->_R^Omega(1) c4(MIN(gen_0':s8_11(n426_11), gen_0':s8_11(n426_11))) ->_IH c4(gen_c2:c3:c49_11(c427_11)) 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: -'(z0, 0') -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(z0, 0') -> c2 MIN(0', z0) -> c3 MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(0') -> c5 TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0') -> 0' twice(s(z0)) -> s(s(twice(z0))) f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1)))) f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1)))) Types: -' :: 0':s -> 0':s -> c:c1 0' :: 0':s c :: c:c1 s :: 0':s -> 0':s c1 :: c:c1 -> c:c1 MIN :: 0':s -> 0':s -> c2:c3:c4 c2 :: c2:c3:c4 c3 :: c2:c3:c4 c4 :: c2:c3:c4 -> c2:c3:c4 TWICE :: 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 F :: 0':s -> 0':s -> c7:c8:c9:c10 c7 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10 - :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s twice :: 0':s -> 0':s c8 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10 c9 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10 c10 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10 f :: 0':s -> 0':s -> f hole_c:c11_11 :: c:c1 hole_0':s2_11 :: 0':s hole_c2:c3:c43_11 :: c2:c3:c4 hole_c5:c64_11 :: c5:c6 hole_c7:c8:c9:c105_11 :: c7:c8:c9:c10 hole_f6_11 :: f gen_c:c17_11 :: Nat -> c:c1 gen_0':s8_11 :: Nat -> 0':s gen_c2:c3:c49_11 :: Nat -> c2:c3:c4 gen_c5:c610_11 :: Nat -> c5:c6 gen_c7:c8:c9:c1011_11 :: Nat -> c7:c8:c9:c10 Lemmas: -'(gen_0':s8_11(n13_11), gen_0':s8_11(n13_11)) -> gen_c:c17_11(n13_11), rt in Omega(1 + n13_11) MIN(gen_0':s8_11(n426_11), gen_0':s8_11(n426_11)) -> gen_c2:c3:c49_11(n426_11), rt in Omega(1 + n426_11) Generator Equations: gen_c:c17_11(0) <=> c gen_c:c17_11(+(x, 1)) <=> c1(gen_c:c17_11(x)) gen_0':s8_11(0) <=> 0' gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x)) gen_c2:c3:c49_11(0) <=> c2 gen_c2:c3:c49_11(+(x, 1)) <=> c4(gen_c2:c3:c49_11(x)) gen_c5:c610_11(0) <=> c5 gen_c5:c610_11(+(x, 1)) <=> c6(gen_c5:c610_11(x)) gen_c7:c8:c9:c1011_11(0) <=> hole_c7:c8:c9:c105_11 gen_c7:c8:c9:c1011_11(+(x, 1)) <=> c7(gen_c7:c8:c9:c1011_11(x), c, c2) The following defined symbols remain to be analysed: TWICE, F, -, min, twice, f They will be analysed ascendingly in the following order: TWICE < F - < F min < F twice < F - < f min < f twice < f ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: TWICE(gen_0':s8_11(n945_11)) -> gen_c5:c610_11(n945_11), rt in Omega(1 + n945_11) Induction Base: TWICE(gen_0':s8_11(0)) ->_R^Omega(1) c5 Induction Step: TWICE(gen_0':s8_11(+(n945_11, 1))) ->_R^Omega(1) c6(TWICE(gen_0':s8_11(n945_11))) ->_IH c6(gen_c5:c610_11(c946_11)) 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: -'(z0, 0') -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(z0, 0') -> c2 MIN(0', z0) -> c3 MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(0') -> c5 TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0') -> 0' twice(s(z0)) -> s(s(twice(z0))) f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1)))) f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1)))) Types: -' :: 0':s -> 0':s -> c:c1 0' :: 0':s c :: c:c1 s :: 0':s -> 0':s c1 :: c:c1 -> c:c1 MIN :: 0':s -> 0':s -> c2:c3:c4 c2 :: c2:c3:c4 c3 :: c2:c3:c4 c4 :: c2:c3:c4 -> c2:c3:c4 TWICE :: 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 F :: 0':s -> 0':s -> c7:c8:c9:c10 c7 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10 - :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s twice :: 0':s -> 0':s c8 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10 c9 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10 c10 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10 f :: 0':s -> 0':s -> f hole_c:c11_11 :: c:c1 hole_0':s2_11 :: 0':s hole_c2:c3:c43_11 :: c2:c3:c4 hole_c5:c64_11 :: c5:c6 hole_c7:c8:c9:c105_11 :: c7:c8:c9:c10 hole_f6_11 :: f gen_c:c17_11 :: Nat -> c:c1 gen_0':s8_11 :: Nat -> 0':s gen_c2:c3:c49_11 :: Nat -> c2:c3:c4 gen_c5:c610_11 :: Nat -> c5:c6 gen_c7:c8:c9:c1011_11 :: Nat -> c7:c8:c9:c10 Lemmas: -'(gen_0':s8_11(n13_11), gen_0':s8_11(n13_11)) -> gen_c:c17_11(n13_11), rt in Omega(1 + n13_11) MIN(gen_0':s8_11(n426_11), gen_0':s8_11(n426_11)) -> gen_c2:c3:c49_11(n426_11), rt in Omega(1 + n426_11) TWICE(gen_0':s8_11(n945_11)) -> gen_c5:c610_11(n945_11), rt in Omega(1 + n945_11) Generator Equations: gen_c:c17_11(0) <=> c gen_c:c17_11(+(x, 1)) <=> c1(gen_c:c17_11(x)) gen_0':s8_11(0) <=> 0' gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x)) gen_c2:c3:c49_11(0) <=> c2 gen_c2:c3:c49_11(+(x, 1)) <=> c4(gen_c2:c3:c49_11(x)) gen_c5:c610_11(0) <=> c5 gen_c5:c610_11(+(x, 1)) <=> c6(gen_c5:c610_11(x)) gen_c7:c8:c9:c1011_11(0) <=> hole_c7:c8:c9:c105_11 gen_c7:c8:c9:c1011_11(+(x, 1)) <=> c7(gen_c7:c8:c9:c1011_11(x), c, c2) The following defined symbols remain to be analysed: -, F, min, twice, f They will be analysed ascendingly in the following order: - < F min < F twice < F - < f min < f twice < f ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: -(gen_0':s8_11(n1336_11), gen_0':s8_11(n1336_11)) -> gen_0':s8_11(0), rt in Omega(0) Induction Base: -(gen_0':s8_11(0), gen_0':s8_11(0)) ->_R^Omega(0) gen_0':s8_11(0) Induction Step: -(gen_0':s8_11(+(n1336_11, 1)), gen_0':s8_11(+(n1336_11, 1))) ->_R^Omega(0) -(gen_0':s8_11(n1336_11), gen_0':s8_11(n1336_11)) ->_IH gen_0':s8_11(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: -'(z0, 0') -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(z0, 0') -> c2 MIN(0', z0) -> c3 MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(0') -> c5 TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0') -> 0' twice(s(z0)) -> s(s(twice(z0))) f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1)))) f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1)))) Types: -' :: 0':s -> 0':s -> c:c1 0' :: 0':s c :: c:c1 s :: 0':s -> 0':s c1 :: c:c1 -> c:c1 MIN :: 0':s -> 0':s -> c2:c3:c4 c2 :: c2:c3:c4 c3 :: c2:c3:c4 c4 :: c2:c3:c4 -> c2:c3:c4 TWICE :: 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 F :: 0':s -> 0':s -> c7:c8:c9:c10 c7 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10 - :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s twice :: 0':s -> 0':s c8 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10 c9 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10 c10 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10 f :: 0':s -> 0':s -> f hole_c:c11_11 :: c:c1 hole_0':s2_11 :: 0':s hole_c2:c3:c43_11 :: c2:c3:c4 hole_c5:c64_11 :: c5:c6 hole_c7:c8:c9:c105_11 :: c7:c8:c9:c10 hole_f6_11 :: f gen_c:c17_11 :: Nat -> c:c1 gen_0':s8_11 :: Nat -> 0':s gen_c2:c3:c49_11 :: Nat -> c2:c3:c4 gen_c5:c610_11 :: Nat -> c5:c6 gen_c7:c8:c9:c1011_11 :: Nat -> c7:c8:c9:c10 Lemmas: -'(gen_0':s8_11(n13_11), gen_0':s8_11(n13_11)) -> gen_c:c17_11(n13_11), rt in Omega(1 + n13_11) MIN(gen_0':s8_11(n426_11), gen_0':s8_11(n426_11)) -> gen_c2:c3:c49_11(n426_11), rt in Omega(1 + n426_11) TWICE(gen_0':s8_11(n945_11)) -> gen_c5:c610_11(n945_11), rt in Omega(1 + n945_11) -(gen_0':s8_11(n1336_11), gen_0':s8_11(n1336_11)) -> gen_0':s8_11(0), rt in Omega(0) Generator Equations: gen_c:c17_11(0) <=> c gen_c:c17_11(+(x, 1)) <=> c1(gen_c:c17_11(x)) gen_0':s8_11(0) <=> 0' gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x)) gen_c2:c3:c49_11(0) <=> c2 gen_c2:c3:c49_11(+(x, 1)) <=> c4(gen_c2:c3:c49_11(x)) gen_c5:c610_11(0) <=> c5 gen_c5:c610_11(+(x, 1)) <=> c6(gen_c5:c610_11(x)) gen_c7:c8:c9:c1011_11(0) <=> hole_c7:c8:c9:c105_11 gen_c7:c8:c9:c1011_11(+(x, 1)) <=> c7(gen_c7:c8:c9:c1011_11(x), c, c2) The following defined symbols remain to be analysed: min, F, twice, f They will be analysed ascendingly in the following order: min < F twice < F min < f twice < f ---------------------------------------- (27) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: min(gen_0':s8_11(n2044_11), gen_0':s8_11(n2044_11)) -> gen_0':s8_11(n2044_11), rt in Omega(0) Induction Base: min(gen_0':s8_11(0), gen_0':s8_11(0)) ->_R^Omega(0) 0' Induction Step: min(gen_0':s8_11(+(n2044_11, 1)), gen_0':s8_11(+(n2044_11, 1))) ->_R^Omega(0) s(min(gen_0':s8_11(n2044_11), gen_0':s8_11(n2044_11))) ->_IH s(gen_0':s8_11(c2045_11)) 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: -'(z0, 0') -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(z0, 0') -> c2 MIN(0', z0) -> c3 MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(0') -> c5 TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0') -> 0' twice(s(z0)) -> s(s(twice(z0))) f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1)))) f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1)))) Types: -' :: 0':s -> 0':s -> c:c1 0' :: 0':s c :: c:c1 s :: 0':s -> 0':s c1 :: c:c1 -> c:c1 MIN :: 0':s -> 0':s -> c2:c3:c4 c2 :: c2:c3:c4 c3 :: c2:c3:c4 c4 :: c2:c3:c4 -> c2:c3:c4 TWICE :: 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 F :: 0':s -> 0':s -> c7:c8:c9:c10 c7 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10 - :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s twice :: 0':s -> 0':s c8 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10 c9 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10 c10 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10 f :: 0':s -> 0':s -> f hole_c:c11_11 :: c:c1 hole_0':s2_11 :: 0':s hole_c2:c3:c43_11 :: c2:c3:c4 hole_c5:c64_11 :: c5:c6 hole_c7:c8:c9:c105_11 :: c7:c8:c9:c10 hole_f6_11 :: f gen_c:c17_11 :: Nat -> c:c1 gen_0':s8_11 :: Nat -> 0':s gen_c2:c3:c49_11 :: Nat -> c2:c3:c4 gen_c5:c610_11 :: Nat -> c5:c6 gen_c7:c8:c9:c1011_11 :: Nat -> c7:c8:c9:c10 Lemmas: -'(gen_0':s8_11(n13_11), gen_0':s8_11(n13_11)) -> gen_c:c17_11(n13_11), rt in Omega(1 + n13_11) MIN(gen_0':s8_11(n426_11), gen_0':s8_11(n426_11)) -> gen_c2:c3:c49_11(n426_11), rt in Omega(1 + n426_11) TWICE(gen_0':s8_11(n945_11)) -> gen_c5:c610_11(n945_11), rt in Omega(1 + n945_11) -(gen_0':s8_11(n1336_11), gen_0':s8_11(n1336_11)) -> gen_0':s8_11(0), rt in Omega(0) min(gen_0':s8_11(n2044_11), gen_0':s8_11(n2044_11)) -> gen_0':s8_11(n2044_11), rt in Omega(0) Generator Equations: gen_c:c17_11(0) <=> c gen_c:c17_11(+(x, 1)) <=> c1(gen_c:c17_11(x)) gen_0':s8_11(0) <=> 0' gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x)) gen_c2:c3:c49_11(0) <=> c2 gen_c2:c3:c49_11(+(x, 1)) <=> c4(gen_c2:c3:c49_11(x)) gen_c5:c610_11(0) <=> c5 gen_c5:c610_11(+(x, 1)) <=> c6(gen_c5:c610_11(x)) gen_c7:c8:c9:c1011_11(0) <=> hole_c7:c8:c9:c105_11 gen_c7:c8:c9:c1011_11(+(x, 1)) <=> c7(gen_c7:c8:c9:c1011_11(x), c, c2) The following defined symbols remain to be analysed: twice, F, f They will be analysed ascendingly in the following order: twice < F twice < f ---------------------------------------- (29) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: twice(gen_0':s8_11(n2554_11)) -> gen_0':s8_11(*(2, n2554_11)), rt in Omega(0) Induction Base: twice(gen_0':s8_11(0)) ->_R^Omega(0) 0' Induction Step: twice(gen_0':s8_11(+(n2554_11, 1))) ->_R^Omega(0) s(s(twice(gen_0':s8_11(n2554_11)))) ->_IH s(s(gen_0':s8_11(*(2, c2555_11)))) 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: -'(z0, 0') -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(z0, 0') -> c2 MIN(0', z0) -> c3 MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(0') -> c5 TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) -(z0, 0') -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0') -> 0' min(0', z0) -> 0' min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0') -> 0' twice(s(z0)) -> s(s(twice(z0))) f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1)))) f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1)))) Types: -' :: 0':s -> 0':s -> c:c1 0' :: 0':s c :: c:c1 s :: 0':s -> 0':s c1 :: c:c1 -> c:c1 MIN :: 0':s -> 0':s -> c2:c3:c4 c2 :: c2:c3:c4 c3 :: c2:c3:c4 c4 :: c2:c3:c4 -> c2:c3:c4 TWICE :: 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 F :: 0':s -> 0':s -> c7:c8:c9:c10 c7 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10 - :: 0':s -> 0':s -> 0':s min :: 0':s -> 0':s -> 0':s twice :: 0':s -> 0':s c8 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10 c9 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10 c10 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10 f :: 0':s -> 0':s -> f hole_c:c11_11 :: c:c1 hole_0':s2_11 :: 0':s hole_c2:c3:c43_11 :: c2:c3:c4 hole_c5:c64_11 :: c5:c6 hole_c7:c8:c9:c105_11 :: c7:c8:c9:c10 hole_f6_11 :: f gen_c:c17_11 :: Nat -> c:c1 gen_0':s8_11 :: Nat -> 0':s gen_c2:c3:c49_11 :: Nat -> c2:c3:c4 gen_c5:c610_11 :: Nat -> c5:c6 gen_c7:c8:c9:c1011_11 :: Nat -> c7:c8:c9:c10 Lemmas: -'(gen_0':s8_11(n13_11), gen_0':s8_11(n13_11)) -> gen_c:c17_11(n13_11), rt in Omega(1 + n13_11) MIN(gen_0':s8_11(n426_11), gen_0':s8_11(n426_11)) -> gen_c2:c3:c49_11(n426_11), rt in Omega(1 + n426_11) TWICE(gen_0':s8_11(n945_11)) -> gen_c5:c610_11(n945_11), rt in Omega(1 + n945_11) -(gen_0':s8_11(n1336_11), gen_0':s8_11(n1336_11)) -> gen_0':s8_11(0), rt in Omega(0) min(gen_0':s8_11(n2044_11), gen_0':s8_11(n2044_11)) -> gen_0':s8_11(n2044_11), rt in Omega(0) twice(gen_0':s8_11(n2554_11)) -> gen_0':s8_11(*(2, n2554_11)), rt in Omega(0) Generator Equations: gen_c:c17_11(0) <=> c gen_c:c17_11(+(x, 1)) <=> c1(gen_c:c17_11(x)) gen_0':s8_11(0) <=> 0' gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x)) gen_c2:c3:c49_11(0) <=> c2 gen_c2:c3:c49_11(+(x, 1)) <=> c4(gen_c2:c3:c49_11(x)) gen_c5:c610_11(0) <=> c5 gen_c5:c610_11(+(x, 1)) <=> c6(gen_c5:c610_11(x)) gen_c7:c8:c9:c1011_11(0) <=> hole_c7:c8:c9:c105_11 gen_c7:c8:c9:c1011_11(+(x, 1)) <=> c7(gen_c7:c8:c9:c1011_11(x), c, c2) The following defined symbols remain to be analysed: F, f ---------------------------------------- (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: -(x, 0) -> x [1] -(s(x), s(y)) -> -(x, y) [1] min(x, 0) -> 0 [1] min(0, y) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y)))) [1] f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y)))) [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: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] min(x, 0) -> 0 [1] min(0, y) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] f(s(x), s(y)) -> f(minus(y, min(x, y)), s(twice(min(x, y)))) [1] f(s(x), s(y)) -> f(minus(x, min(x, y)), s(twice(min(x, y)))) [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: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] min(x, 0) -> 0 [1] min(0, y) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] f(s(x), s(y)) -> f(minus(y, min(x, y)), s(twice(min(x, y)))) [1] f(s(x), s(y)) -> f(minus(x, min(x, y)), s(twice(min(x, y)))) [1] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s min :: 0:s -> 0:s -> 0:s twice :: 0:s -> 0:s f :: 0:s -> 0:s -> f 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: f_2 (c) The following functions are completely defined: minus_2 min_2 twice_1 Due to the following rules being added: minus(v0, v1) -> 0 [0] And the following fresh constants: const ---------------------------------------- (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: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] min(x, 0) -> 0 [1] min(0, y) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] f(s(x), s(y)) -> f(minus(y, min(x, y)), s(twice(min(x, y)))) [1] f(s(x), s(y)) -> f(minus(x, min(x, y)), s(twice(min(x, y)))) [1] minus(v0, v1) -> 0 [0] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s min :: 0:s -> 0:s -> 0:s twice :: 0:s -> 0:s f :: 0:s -> 0:s -> f const :: f 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: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] min(x, 0) -> 0 [1] min(0, y) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] f(s(x), s(0)) -> f(minus(0, 0), s(twice(0))) [3] f(s(0), s(0)) -> f(minus(0, 0), s(twice(0))) [3] f(s(0), s(0)) -> f(minus(0, 0), s(twice(0))) [3] f(s(0), s(y)) -> f(minus(y, 0), s(twice(0))) [3] f(s(s(x')), s(s(y'))) -> f(minus(s(y'), s(min(x', y'))), s(twice(s(min(x', y'))))) [3] f(s(x), s(0)) -> f(minus(x, 0), s(twice(0))) [3] f(s(0), s(0)) -> f(minus(0, 0), s(twice(0))) [3] f(s(0), s(0)) -> f(minus(0, 0), s(twice(0))) [3] f(s(0), s(y)) -> f(minus(0, 0), s(twice(0))) [3] f(s(s(x'')), s(s(y''))) -> f(minus(s(x''), s(min(x'', y''))), s(twice(s(min(x'', y''))))) [3] minus(v0, v1) -> 0 [0] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s min :: 0:s -> 0:s -> 0:s twice :: 0:s -> 0:s f :: 0:s -> 0:s -> f const :: f 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 const => 0 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> f(minus(x, 0), 1 + twice(0)) :|: x >= 0, z' = 1 + 0, z = 1 + x f(z, z') -{ 3 }-> f(minus(y, 0), 1 + twice(0)) :|: z' = 1 + y, z = 1 + 0, y >= 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: x >= 0, z' = 1 + 0, z = 1 + x f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z' = 1 + y, z = 1 + 0, y >= 0 f(z, z') -{ 3 }-> f(minus(1 + x'', 1 + min(x'', y'')), 1 + twice(1 + min(x'', y''))) :|: z' = 1 + (1 + y''), y'' >= 0, x'' >= 0, z = 1 + (1 + x'') f(z, z') -{ 3 }-> f(minus(1 + y', 1 + min(x', y')), 1 + twice(1 + min(x', y'))) :|: z' = 1 + (1 + y'), x' >= 0, y' >= 0, z = 1 + (1 + x') min(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 min(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y min(z, z') -{ 1 }-> 1 + min(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(x)) :|: x >= 0, z = 1 + x ---------------------------------------- (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: f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(z - 1, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(z' - 1, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 ---------------------------------------- (45) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { twice } { minus } { min } { f } ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(z - 1, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(z' - 1, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {twice}, {minus}, {min}, {f} ---------------------------------------- (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: f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(z - 1, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(z' - 1, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {twice}, {minus}, {min}, {f} ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: twice after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(z - 1, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(z' - 1, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {twice}, {minus}, {min}, {f} Previous analysis results are: twice: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: twice after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(z - 1, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(z' - 1, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {minus}, {min}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*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: f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s') :|: s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s'') :|: s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s3) :|: s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 4 }-> f(minus(z - 1, 0), 1 + s2) :|: s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(z' - 1, 0), 1 + s1) :|: s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {minus}, {min}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*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: f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s') :|: s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s'') :|: s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s3) :|: s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 4 }-> f(minus(z - 1, 0), 1 + s2) :|: s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(z' - 1, 0), 1 + s1) :|: s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {minus}, {min}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*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: f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s') :|: s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s'') :|: s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s3) :|: s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 4 }-> f(minus(z - 1, 0), 1 + s2) :|: s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(z' - 1, 0), 1 + s1) :|: s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {min}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*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: f(z, z') -{ 5 }-> f(s5, 1 + s') :|: s5 >= 0, s5 <= 0, s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s6, 1 + s'') :|: s6 >= 0, s6 <= 0, s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s7, 1 + s1) :|: s7 >= 0, s7 <= z' - 1, s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 5 }-> f(s8, 1 + s2) :|: s8 >= 0, s8 <= z - 1, s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s9, 1 + s3) :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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' }-> s4 :|: s4 >= 0, s4 <= 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {min}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*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: f(z, z') -{ 5 }-> f(s5, 1 + s') :|: s5 >= 0, s5 <= 0, s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s6, 1 + s'') :|: s6 >= 0, s6 <= 0, s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s7, 1 + s1) :|: s7 >= 0, s7 <= z' - 1, s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 5 }-> f(s8, 1 + s2) :|: s8 >= 0, s8 <= z - 1, s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s9, 1 + s3) :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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' }-> s4 :|: s4 >= 0, s4 <= 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {min}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*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: f(z, z') -{ 5 }-> f(s5, 1 + s') :|: s5 >= 0, s5 <= 0, s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s6, 1 + s'') :|: s6 >= 0, s6 <= 0, s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s7, 1 + s1) :|: s7 >= 0, s7 <= z' - 1, s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 5 }-> f(s8, 1 + s2) :|: s8 >= 0, s8 <= z - 1, s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s9, 1 + s3) :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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' }-> s4 :|: s4 >= 0, s4 <= 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*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: f(z, z') -{ 7 + s11 + s13 + 2*z' }-> f(s12, 1 + s14) :|: s11 >= 0, s11 <= z - 2, s12 >= 0, s12 <= 1 + (z' - 2), s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), z - 2 >= 0, z' - 2 >= 0 f(z, z') -{ 7 + s15 + s17 + 2*z' }-> f(s16, 1 + s18) :|: s15 >= 0, s15 <= z - 2, s16 >= 0, s16 <= 1 + (z - 2), s17 >= 0, s17 <= z - 2, s18 >= 0, s18 <= 2 * (1 + s17), z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 5 }-> f(s5, 1 + s') :|: s5 >= 0, s5 <= 0, s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s6, 1 + s'') :|: s6 >= 0, s6 <= 0, s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s7, 1 + s1) :|: s7 >= 0, s7 <= z' - 1, s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 5 }-> f(s8, 1 + s2) :|: s8 >= 0, s8 <= z - 1, s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s9, 1 + s3) :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= 2 * 0, 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 + s10 :|: s10 >= 0, s10 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s4 :|: s4 >= 0, s4 <= 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*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: f after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 7 + s11 + s13 + 2*z' }-> f(s12, 1 + s14) :|: s11 >= 0, s11 <= z - 2, s12 >= 0, s12 <= 1 + (z' - 2), s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), z - 2 >= 0, z' - 2 >= 0 f(z, z') -{ 7 + s15 + s17 + 2*z' }-> f(s16, 1 + s18) :|: s15 >= 0, s15 <= z - 2, s16 >= 0, s16 <= 1 + (z - 2), s17 >= 0, s17 <= z - 2, s18 >= 0, s18 <= 2 * (1 + s17), z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 5 }-> f(s5, 1 + s') :|: s5 >= 0, s5 <= 0, s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s6, 1 + s'') :|: s6 >= 0, s6 <= 0, s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s7, 1 + s1) :|: s7 >= 0, s7 <= z' - 1, s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 5 }-> f(s8, 1 + s2) :|: s8 >= 0, s8 <= z - 1, s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s9, 1 + s3) :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= 2 * 0, 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 + s10 :|: s10 >= 0, s10 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s4 :|: s4 >= 0, s4 <= 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*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] f: runtime: ?, size: O(1) [0] ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 7 + s11 + s13 + 2*z' }-> f(s12, 1 + s14) :|: s11 >= 0, s11 <= z - 2, s12 >= 0, s12 <= 1 + (z' - 2), s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), z - 2 >= 0, z' - 2 >= 0 f(z, z') -{ 7 + s15 + s17 + 2*z' }-> f(s16, 1 + s18) :|: s15 >= 0, s15 <= z - 2, s16 >= 0, s16 <= 1 + (z - 2), s17 >= 0, s17 <= z - 2, s18 >= 0, s18 <= 2 * (1 + s17), z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 5 }-> f(s5, 1 + s') :|: s5 >= 0, s5 <= 0, s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s6, 1 + s'') :|: s6 >= 0, s6 <= 0, s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s7, 1 + s1) :|: s7 >= 0, s7 <= z' - 1, s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 5 }-> f(s8, 1 + s2) :|: s8 >= 0, s8 <= z - 1, s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s9, 1 + s3) :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= 2 * 0, 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 + s10 :|: s10 >= 0, s10 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s4 :|: s4 >= 0, s4 <= 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*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] f: runtime: INF, size: O(1) [0] ---------------------------------------- (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] f(v0, v1) -> null_f [0] min(v0, v1) -> null_min [0] twice(v0) -> null_twice [0] And the following fresh constants: null_minus, null_f, null_min, null_twice ---------------------------------------- (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: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] min(x, 0) -> 0 [1] min(0, y) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] f(s(x), s(y)) -> f(minus(y, min(x, y)), s(twice(min(x, y)))) [1] f(s(x), s(y)) -> f(minus(x, min(x, y)), s(twice(min(x, y)))) [1] minus(v0, v1) -> null_minus [0] f(v0, v1) -> null_f [0] min(v0, v1) -> null_min [0] twice(v0) -> null_twice [0] The TRS has the following type information: minus :: 0:s:null_minus:null_min:null_twice -> 0:s:null_minus:null_min:null_twice -> 0:s:null_minus:null_min:null_twice 0 :: 0:s:null_minus:null_min:null_twice s :: 0:s:null_minus:null_min:null_twice -> 0:s:null_minus:null_min:null_twice min :: 0:s:null_minus:null_min:null_twice -> 0:s:null_minus:null_min:null_twice -> 0:s:null_minus:null_min:null_twice twice :: 0:s:null_minus:null_min:null_twice -> 0:s:null_minus:null_min:null_twice f :: 0:s:null_minus:null_min:null_twice -> 0:s:null_minus:null_min:null_twice -> null_f null_minus :: 0:s:null_minus:null_min:null_twice null_f :: null_f null_min :: 0:s:null_minus:null_min:null_twice null_twice :: 0:s:null_minus:null_min:null_twice 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_f => 0 null_min => 0 null_twice => 0 ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> f(minus(x, min(x, y)), 1 + twice(min(x, y))) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x f(z, z') -{ 1 }-> f(minus(y, min(x, y)), 1 + twice(min(x, y))) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 min(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 min(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y min(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 min(z, z') -{ 1 }-> 1 + min(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 twice(z) -{ 1 }-> 1 + (1 + twice(x)) :|: x >= 0, z = 1 + x 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: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1)))) f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1)))) Tuples: -'(z0, 0) -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(z0, 0) -> c2 MIN(0, z0) -> c3 MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(0) -> c5 TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) S tuples: -'(z0, 0) -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(z0, 0) -> c2 MIN(0, z0) -> c3 MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(0) -> c5 TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1, f_2 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c, c1_1, c2, c3, c4_1, c5, c6_1, c7_3, c8_3, c9_3, c10_3 ---------------------------------------- (77) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: MIN(z0, 0) -> c2 TWICE(0) -> c5 MIN(0, z0) -> c3 -'(z0, 0) -> c ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1)))) f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1)))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1, f_2 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_3, c8_3, c9_3, c10_3 ---------------------------------------- (79) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1)))) f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1)))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_3, c8_3, c9_3, c10_3 ---------------------------------------- (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: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) The (relative) TRS S consists of the following rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) 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: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) 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: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1] MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) [1] TWICE(s(z0)) -> c6(TWICE(z0)) [1] F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) [1] F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1] F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) [1] F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1] -(z0, 0) -> z0 [0] -(s(z0), s(z1)) -> -(z0, z1) [0] min(z0, 0) -> 0 [0] min(0, z0) -> 0 [0] min(s(z0), s(z1)) -> s(min(z0, z1)) [0] twice(0) -> 0 [0] twice(s(z0)) -> s(s(twice(z0))) [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: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1] MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) [1] TWICE(s(z0)) -> c6(TWICE(z0)) [1] F(s(z0), s(z1)) -> c7(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) [1] F(s(z0), s(z1)) -> c8(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1] F(s(z0), s(z1)) -> c9(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) [1] F(s(z0), s(z1)) -> c10(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1] minus(z0, 0) -> z0 [0] minus(s(z0), s(z1)) -> minus(z0, z1) [0] min(z0, 0) -> 0 [0] min(0, z0) -> 0 [0] min(s(z0), s(z1)) -> s(min(z0, z1)) [0] twice(0) -> 0 [0] twice(s(z0)) -> s(s(twice(z0))) [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: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1] MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) [1] TWICE(s(z0)) -> c6(TWICE(z0)) [1] F(s(z0), s(z1)) -> c7(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) [1] F(s(z0), s(z1)) -> c8(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1] F(s(z0), s(z1)) -> c9(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) [1] F(s(z0), s(z1)) -> c10(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1] minus(z0, 0) -> z0 [0] minus(s(z0), s(z1)) -> minus(z0, z1) [0] min(z0, 0) -> 0 [0] min(0, z0) -> 0 [0] min(s(z0), s(z1)) -> s(min(z0, z1)) [0] twice(0) -> 0 [0] twice(s(z0)) -> s(s(twice(z0))) [0] The TRS has the following type information: -' :: s:0 -> s:0 -> c1 s :: s:0 -> s:0 c1 :: c1 -> c1 MIN :: s:0 -> s:0 -> c4 c4 :: c4 -> c4 TWICE :: s:0 -> c6 c6 :: c6 -> c6 F :: s:0 -> s:0 -> c7:c8:c9:c10 c7 :: c7:c8:c9:c10 -> c1 -> c4 -> c7:c8:c9:c10 minus :: s:0 -> s:0 -> s:0 min :: s:0 -> s:0 -> s:0 twice :: s:0 -> s:0 c8 :: c7:c8:c9:c10 -> c6 -> c4 -> c7:c8:c9:c10 c9 :: c7:c8:c9:c10 -> c1 -> c4 -> c7:c8:c9:c10 c10 :: c7:c8:c9:c10 -> c6 -> c4 -> c7:c8:c9:c10 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: -'_2 MIN_2 TWICE_1 F_2 (c) The following functions are completely defined: minus_2 min_2 twice_1 Due to the following rules being added: minus(v0, v1) -> 0 [0] min(v0, v1) -> 0 [0] twice(v0) -> 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: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1] MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) [1] TWICE(s(z0)) -> c6(TWICE(z0)) [1] F(s(z0), s(z1)) -> c7(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) [1] F(s(z0), s(z1)) -> c8(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1] F(s(z0), s(z1)) -> c9(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) [1] F(s(z0), s(z1)) -> c10(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1] minus(z0, 0) -> z0 [0] minus(s(z0), s(z1)) -> minus(z0, z1) [0] min(z0, 0) -> 0 [0] min(0, z0) -> 0 [0] min(s(z0), s(z1)) -> s(min(z0, z1)) [0] twice(0) -> 0 [0] twice(s(z0)) -> s(s(twice(z0))) [0] minus(v0, v1) -> 0 [0] min(v0, v1) -> 0 [0] twice(v0) -> 0 [0] The TRS has the following type information: -' :: s:0 -> s:0 -> c1 s :: s:0 -> s:0 c1 :: c1 -> c1 MIN :: s:0 -> s:0 -> c4 c4 :: c4 -> c4 TWICE :: s:0 -> c6 c6 :: c6 -> c6 F :: s:0 -> s:0 -> c7:c8:c9:c10 c7 :: c7:c8:c9:c10 -> c1 -> c4 -> c7:c8:c9:c10 minus :: s:0 -> s:0 -> s:0 min :: s:0 -> s:0 -> s:0 twice :: s:0 -> s:0 c8 :: c7:c8:c9:c10 -> c6 -> c4 -> c7:c8:c9:c10 c9 :: c7:c8:c9:c10 -> c1 -> c4 -> c7:c8:c9:c10 c10 :: c7:c8:c9:c10 -> c6 -> c4 -> c7:c8:c9:c10 0 :: s:0 const :: c1 const1 :: c4 const2 :: c6 const3 :: c7:c8:c9:c10 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: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1] MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) [1] TWICE(s(z0)) -> c6(TWICE(z0)) [1] F(s(z0), s(z1)) -> c7(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) [1] F(s(z0), s(z1)) -> c8(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1] F(s(z0), s(z1)) -> c9(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) [1] F(s(z0), s(z1)) -> c10(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1] minus(z0, 0) -> z0 [0] minus(s(z0), s(z1)) -> minus(z0, z1) [0] min(z0, 0) -> 0 [0] min(0, z0) -> 0 [0] min(s(z0), s(z1)) -> s(min(z0, z1)) [0] twice(0) -> 0 [0] twice(s(z0)) -> s(s(twice(z0))) [0] minus(v0, v1) -> 0 [0] min(v0, v1) -> 0 [0] twice(v0) -> 0 [0] The TRS has the following type information: -' :: s:0 -> s:0 -> c1 s :: s:0 -> s:0 c1 :: c1 -> c1 MIN :: s:0 -> s:0 -> c4 c4 :: c4 -> c4 TWICE :: s:0 -> c6 c6 :: c6 -> c6 F :: s:0 -> s:0 -> c7:c8:c9:c10 c7 :: c7:c8:c9:c10 -> c1 -> c4 -> c7:c8:c9:c10 minus :: s:0 -> s:0 -> s:0 min :: s:0 -> s:0 -> s:0 twice :: s:0 -> s:0 c8 :: c7:c8:c9:c10 -> c6 -> c4 -> c7:c8:c9:c10 c9 :: c7:c8:c9:c10 -> c1 -> c4 -> c7:c8:c9:c10 c10 :: c7:c8:c9:c10 -> c6 -> c4 -> c7:c8:c9:c10 0 :: s:0 const :: c1 const1 :: c4 const2 :: c6 const3 :: c7:c8:c9:c10 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 F(z, z') -{ 1 }-> 1 + F(minus(z0, min(z0, z1)), 1 + twice(min(z0, z1))) + TWICE(min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 F(z, z') -{ 1 }-> 1 + F(minus(z0, min(z0, z1)), 1 + twice(min(z0, z1))) + -'(z0, min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 F(z, z') -{ 1 }-> 1 + F(minus(z1, min(z0, z1)), 1 + twice(min(z0, z1))) + TWICE(min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 F(z, z') -{ 1 }-> 1 + F(minus(z1, min(z0, z1)), 1 + twice(min(z0, z1))) + -'(z1, min(z0, z1)) + MIN(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 TWICE(z) -{ 1 }-> 1 + TWICE(z0) :|: z = 1 + z0, z0 >= 0 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 twice(z) -{ 0 }-> 1 + (1 + twice(z0)) :|: z = 1 + z0, z0 >= 0 ---------------------------------------- (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 F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + MIN(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 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 ---------------------------------------- (99) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { MIN } { twice } { -' } { minus } { min } { TWICE } { F } ---------------------------------------- (100) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + MIN(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 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {MIN}, {twice}, {-'}, {minus}, {min}, {TWICE}, {F} ---------------------------------------- (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 F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + MIN(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 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {MIN}, {twice}, {-'}, {minus}, {min}, {TWICE}, {F} ---------------------------------------- (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 F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + MIN(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 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {MIN}, {twice}, {-'}, {minus}, {min}, {TWICE}, {F} 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 F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + MIN(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 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {twice}, {-'}, {minus}, {min}, {TWICE}, {F} 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 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {twice}, {-'}, {minus}, {min}, {TWICE}, {F} 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: twice after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z ---------------------------------------- (110) Obligation: Complexity RNTS consisting of the following rules: -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {twice}, {-'}, {minus}, {min}, {TWICE}, {F} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] twice: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (111) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: twice 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 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {-'}, {minus}, {min}, {TWICE}, {F} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] twice: runtime: O(1) [0], size: O(n^1) [2*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 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {-'}, {minus}, {min}, {TWICE}, {F} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] twice: runtime: O(1) [0], size: O(n^1) [2*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 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {-'}, {minus}, {min}, {TWICE}, {F} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] twice: runtime: O(1) [0], size: O(n^1) [2*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 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {minus}, {min}, {TWICE}, {F} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] twice: runtime: O(1) [0], size: O(n^1) [2*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 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {minus}, {min}, {TWICE}, {F} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] twice: runtime: O(1) [0], size: O(n^1) [2*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 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {minus}, {min}, {TWICE}, {F} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] twice: runtime: O(1) [0], size: O(n^1) [2*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 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {min}, {TWICE}, {F} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] twice: runtime: O(1) [0], size: O(n^1) [2*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 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 }-> s5 :|: s5 >= 0, s5 <= 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {min}, {TWICE}, {F} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] twice: runtime: O(1) [0], size: O(n^1) [2*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 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 }-> s5 :|: s5 >= 0, s5 <= 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {min}, {TWICE}, {F} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] twice: runtime: O(1) [0], size: O(n^1) [2*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 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 }-> s5 :|: s5 >= 0, s5 <= 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {TWICE}, {F} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] twice: runtime: O(1) [0], size: O(n^1) [2*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 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(s13, 1 + s15) + TWICE(s16) + s'' :|: s12 >= 0, s12 <= z' - 1, s13 >= 0, s13 <= z' - 1, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= z' - 1, s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ s21 + z' }-> 1 + F(s18, 1 + s20) + s22 + s1 :|: s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z - 1, s19 >= 0, s19 <= z' - 1, s20 >= 0, s20 <= 2 * s19, s21 >= 0, s21 <= z' - 1, s22 >= 0, s22 <= 0, s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(s24, 1 + s26) + TWICE(s27) + s2 :|: s23 >= 0, s23 <= z' - 1, s24 >= 0, s24 <= z - 1, s25 >= 0, s25 <= z' - 1, s26 >= 0, s26 <= 2 * s25, s27 >= 0, s27 <= z' - 1, s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ s10 + z' }-> 1 + F(s7, 1 + s9) + s11 + s' :|: s6 >= 0, s6 <= z' - 1, s7 >= 0, s7 <= z' - 1, s8 >= 0, s8 <= z' - 1, s9 >= 0, s9 <= 2 * s8, s10 >= 0, s10 <= z' - 1, s11 >= 0, s11 <= 0, s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 + s28 :|: s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {TWICE}, {F} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] twice: runtime: O(1) [0], size: O(n^1) [2*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: TWICE 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 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(s13, 1 + s15) + TWICE(s16) + s'' :|: s12 >= 0, s12 <= z' - 1, s13 >= 0, s13 <= z' - 1, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= z' - 1, s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ s21 + z' }-> 1 + F(s18, 1 + s20) + s22 + s1 :|: s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z - 1, s19 >= 0, s19 <= z' - 1, s20 >= 0, s20 <= 2 * s19, s21 >= 0, s21 <= z' - 1, s22 >= 0, s22 <= 0, s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(s24, 1 + s26) + TWICE(s27) + s2 :|: s23 >= 0, s23 <= z' - 1, s24 >= 0, s24 <= z - 1, s25 >= 0, s25 <= z' - 1, s26 >= 0, s26 <= 2 * s25, s27 >= 0, s27 <= z' - 1, s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ s10 + z' }-> 1 + F(s7, 1 + s9) + s11 + s' :|: s6 >= 0, s6 <= z' - 1, s7 >= 0, s7 <= z' - 1, s8 >= 0, s8 <= z' - 1, s9 >= 0, s9 <= 2 * s8, s10 >= 0, s10 <= z' - 1, s11 >= 0, s11 <= 0, s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 + s28 :|: s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {TWICE}, {F} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] twice: runtime: O(1) [0], size: O(n^1) [2*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'] TWICE: runtime: ?, size: O(1) [0] ---------------------------------------- (135) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: TWICE 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 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(s13, 1 + s15) + TWICE(s16) + s'' :|: s12 >= 0, s12 <= z' - 1, s13 >= 0, s13 <= z' - 1, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= z' - 1, s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ s21 + z' }-> 1 + F(s18, 1 + s20) + s22 + s1 :|: s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z - 1, s19 >= 0, s19 <= z' - 1, s20 >= 0, s20 <= 2 * s19, s21 >= 0, s21 <= z' - 1, s22 >= 0, s22 <= 0, s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ z' }-> 1 + F(s24, 1 + s26) + TWICE(s27) + s2 :|: s23 >= 0, s23 <= z' - 1, s24 >= 0, s24 <= z - 1, s25 >= 0, s25 <= z' - 1, s26 >= 0, s26 <= 2 * s25, s27 >= 0, s27 <= z' - 1, s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ s10 + z' }-> 1 + F(s7, 1 + s9) + s11 + s' :|: s6 >= 0, s6 <= z' - 1, s7 >= 0, s7 <= z' - 1, s8 >= 0, s8 <= z' - 1, s9 >= 0, s9 <= 2 * s8, s10 >= 0, s10 <= z' - 1, s11 >= 0, s11 <= 0, s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: 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 + s28 :|: s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {F} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] twice: runtime: O(1) [0], size: O(n^1) [2*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'] TWICE: 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 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ s16 + z' }-> 1 + F(s13, 1 + s15) + s30 + s'' :|: s30 >= 0, s30 <= 0, s12 >= 0, s12 <= z' - 1, s13 >= 0, s13 <= z' - 1, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= z' - 1, s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ s21 + z' }-> 1 + F(s18, 1 + s20) + s22 + s1 :|: s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z - 1, s19 >= 0, s19 <= z' - 1, s20 >= 0, s20 <= 2 * s19, s21 >= 0, s21 <= z' - 1, s22 >= 0, s22 <= 0, s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ s27 + z' }-> 1 + F(s24, 1 + s26) + s31 + s2 :|: s31 >= 0, s31 <= 0, s23 >= 0, s23 <= z' - 1, s24 >= 0, s24 <= z - 1, s25 >= 0, s25 <= z' - 1, s26 >= 0, s26 <= 2 * s25, s27 >= 0, s27 <= z' - 1, s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ s10 + z' }-> 1 + F(s7, 1 + s9) + s11 + s' :|: s6 >= 0, s6 <= z' - 1, s7 >= 0, s7 <= z' - 1, s8 >= 0, s8 <= z' - 1, s9 >= 0, s9 <= 2 * s8, s10 >= 0, s10 <= z' - 1, s11 >= 0, s11 <= 0, s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 TWICE(z) -{ z }-> 1 + s29 :|: s29 >= 0, s29 <= 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 + s28 :|: s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {F} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] twice: runtime: O(1) [0], size: O(n^1) [2*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'] TWICE: runtime: O(n^1) [z], size: O(1) [0] ---------------------------------------- (139) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: F 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 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ s16 + z' }-> 1 + F(s13, 1 + s15) + s30 + s'' :|: s30 >= 0, s30 <= 0, s12 >= 0, s12 <= z' - 1, s13 >= 0, s13 <= z' - 1, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= z' - 1, s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ s21 + z' }-> 1 + F(s18, 1 + s20) + s22 + s1 :|: s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z - 1, s19 >= 0, s19 <= z' - 1, s20 >= 0, s20 <= 2 * s19, s21 >= 0, s21 <= z' - 1, s22 >= 0, s22 <= 0, s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ s27 + z' }-> 1 + F(s24, 1 + s26) + s31 + s2 :|: s31 >= 0, s31 <= 0, s23 >= 0, s23 <= z' - 1, s24 >= 0, s24 <= z - 1, s25 >= 0, s25 <= z' - 1, s26 >= 0, s26 <= 2 * s25, s27 >= 0, s27 <= z' - 1, s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ s10 + z' }-> 1 + F(s7, 1 + s9) + s11 + s' :|: s6 >= 0, s6 <= z' - 1, s7 >= 0, s7 <= z' - 1, s8 >= 0, s8 <= z' - 1, s9 >= 0, s9 <= 2 * s8, s10 >= 0, s10 <= z' - 1, s11 >= 0, s11 <= 0, s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 TWICE(z) -{ z }-> 1 + s29 :|: s29 >= 0, s29 <= 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 + s28 :|: s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {F} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] twice: runtime: O(1) [0], size: O(n^1) [2*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'] TWICE: runtime: O(n^1) [z], size: O(1) [0] F: runtime: ?, size: O(1) [0] ---------------------------------------- (141) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: F 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 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ s16 + z' }-> 1 + F(s13, 1 + s15) + s30 + s'' :|: s30 >= 0, s30 <= 0, s12 >= 0, s12 <= z' - 1, s13 >= 0, s13 <= z' - 1, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= z' - 1, s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ s21 + z' }-> 1 + F(s18, 1 + s20) + s22 + s1 :|: s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z - 1, s19 >= 0, s19 <= z' - 1, s20 >= 0, s20 <= 2 * s19, s21 >= 0, s21 <= z' - 1, s22 >= 0, s22 <= 0, s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ s27 + z' }-> 1 + F(s24, 1 + s26) + s31 + s2 :|: s31 >= 0, s31 <= 0, s23 >= 0, s23 <= z' - 1, s24 >= 0, s24 <= z - 1, s25 >= 0, s25 <= z' - 1, s26 >= 0, s26 <= 2 * s25, s27 >= 0, s27 <= z' - 1, s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0 F(z, z') -{ s10 + z' }-> 1 + F(s7, 1 + s9) + s11 + s' :|: s6 >= 0, s6 <= z' - 1, s7 >= 0, s7 <= z' - 1, s8 >= 0, s8 <= z' - 1, s9 >= 0, s9 <= 2 * s8, s10 >= 0, s10 <= z' - 1, s11 >= 0, s11 <= 0, s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0 MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 TWICE(z) -{ z }-> 1 + s29 :|: s29 >= 0, s29 <= 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 + s28 :|: s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: z >= 0 twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {F} Previous analysis results are: MIN: runtime: O(n^1) [z'], size: O(1) [0] twice: runtime: O(1) [0], size: O(n^1) [2*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'] TWICE: runtime: O(n^1) [z], size: O(1) [0] F: 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] min(v0, v1) -> null_min [0] twice(v0) -> null_twice [0] -'(v0, v1) -> null_-' [0] MIN(v0, v1) -> null_MIN [0] TWICE(v0) -> null_TWICE [0] F(v0, v1) -> null_F [0] And the following fresh constants: null_minus, null_min, null_twice, null_-', null_MIN, null_TWICE, null_F ---------------------------------------- (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: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1] MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) [1] TWICE(s(z0)) -> c6(TWICE(z0)) [1] F(s(z0), s(z1)) -> c7(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) [1] F(s(z0), s(z1)) -> c8(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1] F(s(z0), s(z1)) -> c9(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) [1] F(s(z0), s(z1)) -> c10(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1] minus(z0, 0) -> z0 [0] minus(s(z0), s(z1)) -> minus(z0, z1) [0] min(z0, 0) -> 0 [0] min(0, z0) -> 0 [0] min(s(z0), s(z1)) -> s(min(z0, z1)) [0] twice(0) -> 0 [0] twice(s(z0)) -> s(s(twice(z0))) [0] minus(v0, v1) -> null_minus [0] min(v0, v1) -> null_min [0] twice(v0) -> null_twice [0] -'(v0, v1) -> null_-' [0] MIN(v0, v1) -> null_MIN [0] TWICE(v0) -> null_TWICE [0] F(v0, v1) -> null_F [0] The TRS has the following type information: -' :: s:0:null_minus:null_min:null_twice -> s:0:null_minus:null_min:null_twice -> c1:null_-' s :: s:0:null_minus:null_min:null_twice -> s:0:null_minus:null_min:null_twice c1 :: c1:null_-' -> c1:null_-' MIN :: s:0:null_minus:null_min:null_twice -> s:0:null_minus:null_min:null_twice -> c4:null_MIN c4 :: c4:null_MIN -> c4:null_MIN TWICE :: s:0:null_minus:null_min:null_twice -> c6:null_TWICE c6 :: c6:null_TWICE -> c6:null_TWICE F :: s:0:null_minus:null_min:null_twice -> s:0:null_minus:null_min:null_twice -> c7:c8:c9:c10:null_F c7 :: c7:c8:c9:c10:null_F -> c1:null_-' -> c4:null_MIN -> c7:c8:c9:c10:null_F minus :: s:0:null_minus:null_min:null_twice -> s:0:null_minus:null_min:null_twice -> s:0:null_minus:null_min:null_twice min :: s:0:null_minus:null_min:null_twice -> s:0:null_minus:null_min:null_twice -> s:0:null_minus:null_min:null_twice twice :: s:0:null_minus:null_min:null_twice -> s:0:null_minus:null_min:null_twice c8 :: c7:c8:c9:c10:null_F -> c6:null_TWICE -> c4:null_MIN -> c7:c8:c9:c10:null_F c9 :: c7:c8:c9:c10:null_F -> c1:null_-' -> c4:null_MIN -> c7:c8:c9:c10:null_F c10 :: c7:c8:c9:c10:null_F -> c6:null_TWICE -> c4:null_MIN -> c7:c8:c9:c10:null_F 0 :: s:0:null_minus:null_min:null_twice null_minus :: s:0:null_minus:null_min:null_twice null_min :: s:0:null_minus:null_min:null_twice null_twice :: s:0:null_minus:null_min:null_twice null_-' :: c1:null_-' null_MIN :: c4:null_MIN null_TWICE :: c6:null_TWICE null_F :: c7:c8:c9:c10:null_F 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_min => 0 null_twice => 0 null_-' => 0 null_MIN => 0 null_TWICE => 0 null_F => 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 F(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 F(z, z') -{ 1 }-> 1 + F(minus(z0, min(z0, z1)), 1 + twice(min(z0, z1))) + TWICE(min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 F(z, z') -{ 1 }-> 1 + F(minus(z0, min(z0, z1)), 1 + twice(min(z0, z1))) + -'(z0, min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 F(z, z') -{ 1 }-> 1 + F(minus(z1, min(z0, z1)), 1 + twice(min(z0, z1))) + TWICE(min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 F(z, z') -{ 1 }-> 1 + F(minus(z1, min(z0, z1)), 1 + twice(min(z0, z1))) + -'(z1, min(z0, z1)) + MIN(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 TWICE(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 TWICE(z) -{ 1 }-> 1 + TWICE(z0) :|: z = 1 + z0, z0 >= 0 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 twice(z) -{ 0 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 twice(z) -{ 0 }-> 1 + (1 + twice(z0)) :|: z = 1 + z0, z0 >= 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (147) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) by F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))), -'(0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))), -'(0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))), -'(0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))), -'(0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))), -'(0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))), -'(0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_3, c9_3, c10_3, c7_3 ---------------------------------------- (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) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_3, c9_3, c10_3, c7_3, c7_1, c7_2 ---------------------------------------- (151) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) by F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_3, c10_3, c7_3, c7_1, c7_2, c8_3 ---------------------------------------- (153) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_3, c10_3, c7_3, c7_1, c7_2, c8_3, c8_2 ---------------------------------------- (155) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) by F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))), -'(0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))), -'(0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))), -'(0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))), -'(0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))), -'(0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))), -'(0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c10_3, c7_3, c7_1, c7_2, c8_3, c8_2, c9_3 ---------------------------------------- (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) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c10_3, c7_3, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1 ---------------------------------------- (159) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) by F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3 ---------------------------------------- (161) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2 ---------------------------------------- (163) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2 ---------------------------------------- (165) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3 ---------------------------------------- (167) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) by F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3 ---------------------------------------- (169) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) by F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3 ---------------------------------------- (171) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3 ---------------------------------------- (173) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) by F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(x0), s(0)) -> c7(F(0, s(twice(min(x0, 0))))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(x0), s(0)) -> c7(F(0, s(twice(min(x0, 0))))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(x0), s(0)) -> c7(F(0, s(twice(min(x0, 0))))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1 ---------------------------------------- (175) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(x0), s(0)) -> c7(F(0, s(twice(min(x0, 0))))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1 ---------------------------------------- (177) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) by F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2 ---------------------------------------- (179) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2 ---------------------------------------- (181) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2 ---------------------------------------- (183) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3 ---------------------------------------- (185) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) by F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3 ---------------------------------------- (187) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) by F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3 ---------------------------------------- (189) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) by F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(x0), s(0)) -> c8(F(0, s(twice(min(x0, 0)))), TWICE(min(x0, 0))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(F(0, s(twice(min(x0, 0)))), TWICE(min(x0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(F(0, s(twice(min(x0, 0)))), TWICE(min(x0, 0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3 ---------------------------------------- (191) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_1 ---------------------------------------- (193) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) We considered the (Usable) Rules:none And the Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1, x_2)) = [1] POL(MIN(x_1, x_2)) = 0 POL(TWICE(x_1)) = 0 POL(c1(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(min(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_1 ---------------------------------------- (195) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) by F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1 ---------------------------------------- (197) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1 ---------------------------------------- (199) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3 ---------------------------------------- (201) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) by F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_1, c9_2, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3 ---------------------------------------- (203) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_1, c9_2, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3 ---------------------------------------- (205) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) by F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3 ---------------------------------------- (207) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) by F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2 ---------------------------------------- (209) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2 ---------------------------------------- (211) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) by F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(0, s(twice(min(0, x0))))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(x0)) -> c9(F(0, s(twice(min(0, x0))))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(x0)) -> c9(F(0, s(twice(min(0, x0))))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1 ---------------------------------------- (213) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(0), s(x0)) -> c9(F(0, s(twice(min(0, x0))))) ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1 ---------------------------------------- (215) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1 ---------------------------------------- (217) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3 ---------------------------------------- (219) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) by F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3 ---------------------------------------- (221) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) by F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3 ---------------------------------------- (223) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) by F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) ---------------------------------------- (224) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3 ---------------------------------------- (225) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) by F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(x0)) -> c10(F(0, s(twice(min(0, x0)))), TWICE(min(0, x0))) ---------------------------------------- (226) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(F(0, s(twice(min(0, x0)))), TWICE(min(0, x0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(F(0, s(twice(min(0, x0)))), TWICE(min(0, x0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2 ---------------------------------------- (227) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (228) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1 ---------------------------------------- (229) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) We considered the (Usable) Rules:none And the Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1, x_2)) = [1] POL(MIN(x_1, x_2)) = 0 POL(TWICE(x_1)) = 0 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(min(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(twice(x_1)) = 0 ---------------------------------------- (230) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1 ---------------------------------------- (231) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace -'(s(z0), s(z1)) -> c1(-'(z0, z1)) by -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) ---------------------------------------- (232) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) S tuples: MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: MIN_2, TWICE_1, F_2, -'_2 Compound Symbols: c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1 ---------------------------------------- (233) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (234) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), MIN(s(0), s(z0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), MIN(s(0), s(z0))) S tuples: MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), MIN(s(0), s(z0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), MIN(s(0), s(z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: MIN_2, TWICE_1, F_2, -'_2 Compound Symbols: c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1 ---------------------------------------- (235) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) by MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) ---------------------------------------- (236) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), MIN(s(0), s(z0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), MIN(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) S tuples: TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), MIN(s(0), s(z0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), MIN(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: TWICE_1, F_2, -'_2, MIN_2 Compound Symbols: c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1 ---------------------------------------- (237) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 16 trailing tuple parts ---------------------------------------- (238) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) S tuples: TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: TWICE_1, F_2, -'_2, MIN_2 Compound Symbols: c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1 ---------------------------------------- (239) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace TWICE(s(z0)) -> c6(TWICE(z0)) by TWICE(s(s(y0))) -> c6(TWICE(s(y0))) ---------------------------------------- (240) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (241) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) by F(s(z0), s(0)) -> c7(F(-(0, 0), s(0))) ---------------------------------------- (242) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (243) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c7(F(-(0, 0), s(0))) by F(s(x0), s(0)) -> c7(F(0, s(0))) ---------------------------------------- (244) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(x0), s(0)) -> c7(F(0, s(0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(x0), s(0)) -> c7(F(0, s(0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (245) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (246) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(x0), s(0)) -> c7 S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(x0), s(0)) -> c7 K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1, c7 ---------------------------------------- (247) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0), s(0)) -> c7 We considered the (Usable) Rules:none And the Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(x0), s(0)) -> c7 The order we found is given by the following interpretation: Polynomial interpretation : POL(-(x_1, x_2)) = [1] + x_1 POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1, x_2)) = [1] + x_2 POL(MIN(x_1, x_2)) = x_1 + x_2 POL(TWICE(x_1)) = 0 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7) = 0 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(min(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] ---------------------------------------- (248) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(x0), s(0)) -> c7 S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) F(s(x0), s(0)) -> c7 Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1, c7 ---------------------------------------- (249) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) by F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) ---------------------------------------- (250) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(x0), s(0)) -> c7 F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) F(s(x0), s(0)) -> c7 Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1, c7 ---------------------------------------- (251) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(x0), s(0)) -> c7 ---------------------------------------- (252) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (253) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) by F(s(0), s(0)) -> c7(F(0, s(0))) ---------------------------------------- (254) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7(F(0, s(0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7(F(0, s(0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (255) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (256) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7 S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7 K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1, c7 ---------------------------------------- (257) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(0), s(0)) -> c7 We considered the (Usable) Rules:none And the Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7 The order we found is given by the following interpretation: Polynomial interpretation : POL(-(x_1, x_2)) = [1] + x_1 POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1, x_2)) = [1] + x_2 POL(MIN(x_1, x_2)) = x_1 + x_2 POL(TWICE(x_1)) = 0 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7) = 0 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(min(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] ---------------------------------------- (258) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7 S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) F(s(0), s(0)) -> c7 Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1, c7 ---------------------------------------- (259) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) by F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) ---------------------------------------- (260) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7 F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) F(s(0), s(0)) -> c7 Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1, c7 ---------------------------------------- (261) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(0), s(0)) -> c7 ---------------------------------------- (262) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (263) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) by F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) ---------------------------------------- (264) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (265) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) by F(s(0), s(0)) -> c7(F(0, s(0))) ---------------------------------------- (266) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) F(s(0), s(0)) -> c7(F(0, s(0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (267) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(0), s(0)) -> c7(F(0, s(0))) ---------------------------------------- (268) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (269) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) by F(s(z0), s(0)) -> c7(F(-(0, 0), s(0))) ---------------------------------------- (270) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (271) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c7(F(-(0, 0), s(0))) by F(s(x0), s(0)) -> c7(F(0, s(0))) ---------------------------------------- (272) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) F(s(x0), s(0)) -> c7(F(0, s(0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (273) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(x0), s(0)) -> c7(F(0, s(0))) ---------------------------------------- (274) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (275) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) by F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) ---------------------------------------- (276) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1