KILLED proof of input_YQKRFMwsXi.trs # AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 135 ms] (2) CpxRelTRS (3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CdtProblem (9) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxRelTRS (11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxRelTRS (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (14) typed CpxTrs (15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (16) TRS for Loop Detection (17) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (18) CdtProblem (19) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (20) CdtProblem (21) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CdtProblem (23) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CpxRelTRS (25) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (26) CpxTRS (27) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CpxWeightedTrs (29) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CpxWeightedTrs (31) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CpxTypedWeightedTrs (33) CompletionProof [UPPER BOUND(ID), 0 ms] (34) CpxTypedWeightedCompleteTrs (35) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CpxTypedWeightedCompleteTrs (37) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 12 ms] (38) CpxRNTS (39) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CpxRNTS (41) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 162 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 73 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 1 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 383 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 63 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 4 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 2136 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 416 ms] (60) CpxRNTS (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 3382 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 544 ms] (66) CpxRNTS (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 9100 ms] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 3025 ms] (72) CpxRNTS (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (74) CpxRNTS (75) IntTrsBoundProof [UPPER BOUND(ID), 239 ms] (76) CpxRNTS (77) IntTrsBoundProof [UPPER BOUND(ID), 32 ms] (78) CpxRNTS (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (80) CpxRNTS (81) IntTrsBoundProof [UPPER BOUND(ID), 26.2 s] (82) CpxRNTS (83) IntTrsBoundProof [UPPER BOUND(ID), 10.3 s] (84) CpxRNTS (85) CompletionProof [UPPER BOUND(ID), 0 ms] (86) CpxTypedWeightedCompleteTrs (87) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (88) CpxRNTS (89) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 528 ms] (90) CdtProblem (91) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 610 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 613 ms] (102) CdtProblem (103) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 454 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 2463 ms] (122) CdtProblem (123) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 3340 ms] (124) CdtProblem (125) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 4532 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (152) CdtProblem (153) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtLeafRemovalProof [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) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (194) CdtProblem (195) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (210) CdtProblem (211) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (212) CdtProblem (213) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (216) CdtProblem (217) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (218) CdtProblem (219) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (220) CdtProblem (221) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (222) CdtProblem (223) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (224) CdtProblem (225) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (226) CdtProblem (227) CdtRewritingProof [BOTH BOUNDS(ID, ID), 4 ms] (228) CdtProblem (229) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (230) CdtProblem (231) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (232) CdtProblem (233) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (234) CdtProblem (235) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (236) CdtProblem (237) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (238) CdtProblem (239) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (240) CdtProblem (241) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (242) CpxWeightedTrs (243) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (244) CpxWeightedTrs (245) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (246) CpxTypedWeightedTrs (247) CompletionProof [UPPER BOUND(ID), 0 ms] (248) CpxTypedWeightedCompleteTrs (249) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (250) CpxTypedWeightedCompleteTrs (251) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 14 ms] (252) CpxRNTS (253) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (254) CpxRNTS (255) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (256) CpxRNTS (257) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (258) CpxRNTS (259) IntTrsBoundProof [UPPER BOUND(ID), 84 ms] (260) CpxRNTS (261) IntTrsBoundProof [UPPER BOUND(ID), 6 ms] (262) CpxRNTS (263) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (264) CpxRNTS (265) IntTrsBoundProof [UPPER BOUND(ID), 358 ms] (266) CpxRNTS (267) IntTrsBoundProof [UPPER BOUND(ID), 108 ms] (268) CpxRNTS (269) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (270) CpxRNTS (271) IntTrsBoundProof [UPPER BOUND(ID), 3198 ms] (272) CpxRNTS (273) IntTrsBoundProof [UPPER BOUND(ID), 1022 ms] (274) CpxRNTS (275) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (276) CpxRNTS (277) IntTrsBoundProof [UPPER BOUND(ID), 6114 ms] (278) CpxRNTS (279) IntTrsBoundProof [UPPER BOUND(ID), 2222 ms] (280) CpxRNTS (281) CompletionProof [UPPER BOUND(ID), 0 ms] (282) CpxTypedWeightedCompleteTrs (283) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (284) CpxRNTS ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil) mergesort(Cons(x, Nil)) -> Cons(x, Nil) merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs)) merge(Cons(x, xs), Nil) -> Cons(x, xs) splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1) splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2)) mergesort(Nil) -> Nil merge(Nil, xs2) -> xs2 notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False goal(xs) -> mergesort(xs) The (relative) TRS S consists of the following rules: <=(S(x), S(y)) -> <=(x, y) <=(0, y) -> True <=(S(x), 0) -> False merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs)) merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2)) Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil) mergesort(Cons(x, Nil)) -> Cons(x, Nil) merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs)) merge(Cons(x, xs), Nil) -> Cons(x, xs) splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1) splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2)) mergesort(Nil) -> Nil merge(Nil, xs2) -> xs2 notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False goal(xs) -> mergesort(xs) The (relative) TRS S consists of the following rules: <=(S(x), S(y)) -> <=(x, y) <=(0, y) -> True <=(S(x), 0) -> False merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs)) merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2)) Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil) mergesort(Cons(x, Nil)) -> Cons(x, Nil) merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs)) merge(Cons(x, xs), Nil) -> Cons(x, xs) splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1) splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2)) mergesort(Nil) -> Nil merge(Nil, xs2) -> xs2 notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False goal(xs) -> mergesort(xs) The (relative) TRS S consists of the following rules: <=(S(x), S(y)) -> <=(x, y) <=(0', y) -> True <=(S(x), 0') -> False merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs)) merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2)) Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) 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: mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil) mergesort(Cons(x, Nil)) -> Cons(x, Nil) merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs)) merge(Cons(x, xs), Nil) -> Cons(x, xs) splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1) splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2)) mergesort(Nil) -> Nil merge(Nil, xs2) -> xs2 notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False goal(xs) -> mergesort(xs) <=(S(x), S(y)) -> <=(x, y) <=(0, y) -> True <=(S(x), 0) -> False merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs)) merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (7) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False goal(z0) -> mergesort(z0) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) <='(0, z0) -> c1 <='(S(z0), 0) -> c2 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGESORT(Cons(z0, Nil)) -> c6 MERGESORT(Nil) -> c7 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) NOTEMPTY(Cons(z0, z1)) -> c14 NOTEMPTY(Nil) -> c15 GOAL(z0) -> c16(MERGESORT(z0)) S tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGESORT(Cons(z0, Nil)) -> c6 MERGESORT(Nil) -> c7 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) NOTEMPTY(Cons(z0, z1)) -> c14 NOTEMPTY(Nil) -> c15 GOAL(z0) -> c16(MERGESORT(z0)) K tuples:none Defined Rule Symbols: mergesort_1, merge_2, splitmerge_3, notEmpty_1, goal_1, <=_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3, NOTEMPTY_1, GOAL_1 Compound Symbols: c_1, c1, c2, c3_1, c4_1, c5_1, c6, c7, c8_2, c9, c10, c11_1, c12_2, c13_2, c14, c15, c16_1 ---------------------------------------- (9) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (10) 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: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGESORT(Cons(z0, Nil)) -> c6 MERGESORT(Nil) -> c7 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) NOTEMPTY(Cons(z0, z1)) -> c14 NOTEMPTY(Nil) -> c15 GOAL(z0) -> c16(MERGESORT(z0)) The (relative) TRS S consists of the following rules: <='(S(z0), S(z1)) -> c(<='(z0, z1)) <='(0, z0) -> c1 <='(S(z0), 0) -> c2 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False goal(z0) -> mergesort(z0) Rewrite Strategy: INNERMOST ---------------------------------------- (11) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (12) 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: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGESORT(Cons(z0, Nil)) -> c6 MERGESORT(Nil) -> c7 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) NOTEMPTY(Cons(z0, z1)) -> c14 NOTEMPTY(Nil) -> c15 GOAL(z0) -> c16(MERGESORT(z0)) The (relative) TRS S consists of the following rules: <='(S(z0), S(z1)) -> c(<='(z0, z1)) <='(0', z0) -> c1 <='(S(z0), 0') -> c2 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) <=(S(z0), S(z1)) -> <=(z0, z1) <=(0', z0) -> True <=(S(z0), 0') -> False merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False goal(z0) -> mergesort(z0) Rewrite Strategy: INNERMOST ---------------------------------------- (13) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (14) Obligation: Innermost TRS: Rules: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGESORT(Cons(z0, Nil)) -> c6 MERGESORT(Nil) -> c7 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) NOTEMPTY(Cons(z0, z1)) -> c14 NOTEMPTY(Nil) -> c15 GOAL(z0) -> c16(MERGESORT(z0)) <='(S(z0), S(z1)) -> c(<='(z0, z1)) <='(0', z0) -> c1 <='(S(z0), 0') -> c2 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) <=(S(z0), S(z1)) -> <=(z0, z1) <=(0', z0) -> True <=(S(z0), 0') -> False merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False goal(z0) -> mergesort(z0) Types: MERGESORT :: Cons:Nil -> c5:c6:c7 Cons :: S:0' -> Cons:Nil -> Cons:Nil c5 :: c11:c12:c13 -> c5:c6:c7 SPLITMERGE :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c11:c12:c13 Nil :: Cons:Nil c6 :: c5:c6:c7 c7 :: c5:c6:c7 MERGE :: Cons:Nil -> Cons:Nil -> c8:c9:c10 c8 :: c3:c4 -> c:c1:c2 -> c8:c9:c10 MERGE[ITE] :: False:True -> Cons:Nil -> Cons:Nil -> c3:c4 <= :: S:0' -> S:0' -> False:True <=' :: S:0' -> S:0' -> c:c1:c2 c9 :: c8:c9:c10 c10 :: c8:c9:c10 c11 :: c11:c12:c13 -> c11:c12:c13 c12 :: c8:c9:c10 -> c5:c6:c7 -> c11:c12:c13 mergesort :: Cons:Nil -> Cons:Nil c13 :: c8:c9:c10 -> c5:c6:c7 -> c11:c12:c13 NOTEMPTY :: Cons:Nil -> c14:c15 c14 :: c14:c15 c15 :: c14:c15 GOAL :: Cons:Nil -> c16 c16 :: c5:c6:c7 -> c16 S :: S:0' -> S:0' c :: c:c1:c2 -> c:c1:c2 0' :: S:0' c1 :: c:c1:c2 c2 :: c:c1:c2 False :: False:True c3 :: c8:c9:c10 -> c3:c4 True :: False:True c4 :: c8:c9:c10 -> c3:c4 merge[Ite] :: False:True -> Cons:Nil -> Cons:Nil -> Cons:Nil merge :: Cons:Nil -> Cons:Nil -> Cons:Nil splitmerge :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil notEmpty :: Cons:Nil -> False:True goal :: Cons:Nil -> Cons:Nil hole_c5:c6:c71_17 :: c5:c6:c7 hole_Cons:Nil2_17 :: Cons:Nil hole_S:0'3_17 :: S:0' hole_c11:c12:c134_17 :: c11:c12:c13 hole_c8:c9:c105_17 :: c8:c9:c10 hole_c3:c46_17 :: c3:c4 hole_c:c1:c27_17 :: c:c1:c2 hole_False:True8_17 :: False:True hole_c14:c159_17 :: c14:c15 hole_c1610_17 :: c16 gen_Cons:Nil11_17 :: Nat -> Cons:Nil gen_S:0'12_17 :: Nat -> S:0' gen_c11:c12:c1313_17 :: Nat -> c11:c12:c13 gen_c:c1:c214_17 :: Nat -> c:c1:c2 ---------------------------------------- (15) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (16) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGESORT(Cons(z0, Nil)) -> c6 MERGESORT(Nil) -> c7 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) NOTEMPTY(Cons(z0, z1)) -> c14 NOTEMPTY(Nil) -> c15 GOAL(z0) -> c16(MERGESORT(z0)) The (relative) TRS S consists of the following rules: <='(S(z0), S(z1)) -> c(<='(z0, z1)) <='(0, z0) -> c1 <='(S(z0), 0) -> c2 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False goal(z0) -> mergesort(z0) Rewrite Strategy: INNERMOST ---------------------------------------- (17) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (18) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False goal(z0) -> mergesort(z0) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) <='(0, z0) -> c1 <='(S(z0), 0) -> c2 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGESORT(Cons(z0, Nil)) -> c6 MERGESORT(Nil) -> c7 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) NOTEMPTY(Cons(z0, z1)) -> c14 NOTEMPTY(Nil) -> c15 GOAL(z0) -> c16(MERGESORT(z0)) S tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGESORT(Cons(z0, Nil)) -> c6 MERGESORT(Nil) -> c7 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) NOTEMPTY(Cons(z0, z1)) -> c14 NOTEMPTY(Nil) -> c15 GOAL(z0) -> c16(MERGESORT(z0)) K tuples:none Defined Rule Symbols: mergesort_1, merge_2, splitmerge_3, notEmpty_1, goal_1, <=_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3, NOTEMPTY_1, GOAL_1 Compound Symbols: c_1, c1, c2, c3_1, c4_1, c5_1, c6, c7, c8_2, c9, c10, c11_1, c12_2, c13_2, c14, c15, c16_1 ---------------------------------------- (19) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: GOAL(z0) -> c16(MERGESORT(z0)) Removed 6 trailing nodes: NOTEMPTY(Cons(z0, z1)) -> c14 <='(S(z0), 0) -> c2 <='(0, z0) -> c1 NOTEMPTY(Nil) -> c15 MERGESORT(Nil) -> c7 MERGESORT(Cons(z0, Nil)) -> c6 ---------------------------------------- (20) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False goal(z0) -> mergesort(z0) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) S tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) K tuples:none Defined Rule Symbols: mergesort_1, merge_2, splitmerge_3, notEmpty_1, goal_1, <=_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3 Compound Symbols: c_1, c3_1, c4_1, c5_1, c8_2, c9, c10, c11_1, c12_2, c13_2 ---------------------------------------- (21) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False goal(z0) -> mergesort(z0) ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) S tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) K tuples:none Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3 Compound Symbols: c_1, c3_1, c4_1, c5_1, c8_2, c9, c10, c11_1, c12_2, c13_2 ---------------------------------------- (23) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (24) 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: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) The (relative) TRS S consists of the following rules: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Rewrite Strategy: INNERMOST ---------------------------------------- (25) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (26) 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: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (27) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (28) 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: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) [1] MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) [1] MERGE(Cons(z0, z1), Nil) -> c9 [1] MERGE(Nil, z0) -> c10 [1] SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) [1] SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) [1] SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) [1] <='(S(z0), S(z1)) -> c(<='(z0, z1)) [0] MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) [0] MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) [0] <=(S(z0), S(z1)) -> <=(z0, z1) [0] <=(0, z0) -> True [0] <=(S(z0), 0) -> False [0] mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) [0] mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) [0] mergesort(Nil) -> Nil [0] splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) [0] splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) [0] merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) [0] merge(Cons(z0, z1), Nil) -> Cons(z0, z1) [0] merge(Nil, z0) -> z0 [0] merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) [0] merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (29) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: <= => lteq ---------------------------------------- (30) 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: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) [1] MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](lteq(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) [1] MERGE(Cons(z0, z1), Nil) -> c9 [1] MERGE(Nil, z0) -> c10 [1] SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) [1] SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) [1] SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) [1] <='(S(z0), S(z1)) -> c(<='(z0, z1)) [0] MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) [0] MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) [0] lteq(S(z0), S(z1)) -> lteq(z0, z1) [0] lteq(0, z0) -> True [0] lteq(S(z0), 0) -> False [0] mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) [0] mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) [0] mergesort(Nil) -> Nil [0] splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) [0] splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) [0] merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](lteq(z0, z2), Cons(z0, z1), Cons(z2, z3)) [0] merge(Cons(z0, z1), Nil) -> Cons(z0, z1) [0] merge(Nil, z0) -> z0 [0] merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) [0] merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (31) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (32) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) [1] MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](lteq(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) [1] MERGE(Cons(z0, z1), Nil) -> c9 [1] MERGE(Nil, z0) -> c10 [1] SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) [1] SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) [1] SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) [1] <='(S(z0), S(z1)) -> c(<='(z0, z1)) [0] MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) [0] MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) [0] lteq(S(z0), S(z1)) -> lteq(z0, z1) [0] lteq(0, z0) -> True [0] lteq(S(z0), 0) -> False [0] mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) [0] mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) [0] mergesort(Nil) -> Nil [0] splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) [0] splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) [0] merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](lteq(z0, z2), Cons(z0, z1), Cons(z2, z3)) [0] merge(Cons(z0, z1), Nil) -> Cons(z0, z1) [0] merge(Nil, z0) -> z0 [0] merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) [0] merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) [0] The TRS has the following type information: MERGESORT :: Cons:Nil -> c5 Cons :: S:0 -> Cons:Nil -> Cons:Nil c5 :: c11:c12:c13 -> c5 SPLITMERGE :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c11:c12:c13 Nil :: Cons:Nil MERGE :: Cons:Nil -> Cons:Nil -> c8:c9:c10 c8 :: c3:c4 -> c -> c8:c9:c10 MERGE[ITE] :: False:True -> Cons:Nil -> Cons:Nil -> c3:c4 lteq :: S:0 -> S:0 -> False:True <=' :: S:0 -> S:0 -> c c9 :: c8:c9:c10 c10 :: c8:c9:c10 c11 :: c11:c12:c13 -> c11:c12:c13 c12 :: c8:c9:c10 -> c5 -> c11:c12:c13 mergesort :: Cons:Nil -> Cons:Nil c13 :: c8:c9:c10 -> c5 -> c11:c12:c13 S :: S:0 -> S:0 c :: c -> c False :: False:True c3 :: c8:c9:c10 -> c3:c4 True :: False:True c4 :: c8:c9:c10 -> c3:c4 0 :: S:0 splitmerge :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil merge :: Cons:Nil -> Cons:Nil -> Cons:Nil merge[Ite] :: False:True -> Cons:Nil -> Cons:Nil -> Cons:Nil Rewrite Strategy: INNERMOST ---------------------------------------- (33) 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: MERGESORT_1 MERGE_2 SPLITMERGE_3 (c) The following functions are completely defined: <='_2 MERGE[ITE]_3 lteq_2 mergesort_1 splitmerge_3 merge_2 merge[Ite]_3 Due to the following rules being added: <='(v0, v1) -> const3 [0] MERGE[ITE](v0, v1, v2) -> const2 [0] lteq(v0, v1) -> null_lteq [0] mergesort(v0) -> Nil [0] splitmerge(v0, v1, v2) -> Nil [0] merge(v0, v1) -> Nil [0] merge[Ite](v0, v1, v2) -> Nil [0] And the following fresh constants: const3, const2, null_lteq, const, const1 ---------------------------------------- (34) 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: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) [1] MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](lteq(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) [1] MERGE(Cons(z0, z1), Nil) -> c9 [1] MERGE(Nil, z0) -> c10 [1] SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) [1] SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) [1] SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) [1] <='(S(z0), S(z1)) -> c(<='(z0, z1)) [0] MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) [0] MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) [0] lteq(S(z0), S(z1)) -> lteq(z0, z1) [0] lteq(0, z0) -> True [0] lteq(S(z0), 0) -> False [0] mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) [0] mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) [0] mergesort(Nil) -> Nil [0] splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) [0] splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) [0] merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](lteq(z0, z2), Cons(z0, z1), Cons(z2, z3)) [0] merge(Cons(z0, z1), Nil) -> Cons(z0, z1) [0] merge(Nil, z0) -> z0 [0] merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) [0] merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) [0] <='(v0, v1) -> const3 [0] MERGE[ITE](v0, v1, v2) -> const2 [0] lteq(v0, v1) -> null_lteq [0] mergesort(v0) -> Nil [0] splitmerge(v0, v1, v2) -> Nil [0] merge(v0, v1) -> Nil [0] merge[Ite](v0, v1, v2) -> Nil [0] The TRS has the following type information: MERGESORT :: Cons:Nil -> c5 Cons :: S:0 -> Cons:Nil -> Cons:Nil c5 :: c11:c12:c13 -> c5 SPLITMERGE :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c11:c12:c13 Nil :: Cons:Nil MERGE :: Cons:Nil -> Cons:Nil -> c8:c9:c10 c8 :: c3:c4:const2 -> c:const3 -> c8:c9:c10 MERGE[ITE] :: False:True:null_lteq -> Cons:Nil -> Cons:Nil -> c3:c4:const2 lteq :: S:0 -> S:0 -> False:True:null_lteq <=' :: S:0 -> S:0 -> c:const3 c9 :: c8:c9:c10 c10 :: c8:c9:c10 c11 :: c11:c12:c13 -> c11:c12:c13 c12 :: c8:c9:c10 -> c5 -> c11:c12:c13 mergesort :: Cons:Nil -> Cons:Nil c13 :: c8:c9:c10 -> c5 -> c11:c12:c13 S :: S:0 -> S:0 c :: c:const3 -> c:const3 False :: False:True:null_lteq c3 :: c8:c9:c10 -> c3:c4:const2 True :: False:True:null_lteq c4 :: c8:c9:c10 -> c3:c4:const2 0 :: S:0 splitmerge :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil merge :: Cons:Nil -> Cons:Nil -> Cons:Nil merge[Ite] :: False:True:null_lteq -> Cons:Nil -> Cons:Nil -> Cons:Nil const3 :: c:const3 const2 :: c3:c4:const2 null_lteq :: False:True:null_lteq const :: c5 const1 :: c11:c12:c13 Rewrite Strategy: INNERMOST ---------------------------------------- (35) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (36) 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: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) [1] MERGE(Cons(S(z0'), z1), Cons(S(z1'), z3)) -> c8(MERGE[ITE](lteq(z0', z1'), Cons(S(z0'), z1), Cons(S(z1'), z3)), <='(S(z0'), S(z1'))) [1] MERGE(Cons(0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](True, Cons(0, z1), Cons(z2, z3)), <='(0, z2)) [1] MERGE(Cons(S(z0''), z1), Cons(0, z3)) -> c8(MERGE[ITE](False, Cons(S(z0''), z1), Cons(0, z3)), <='(S(z0''), 0)) [1] MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](null_lteq, Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) [1] MERGE(Cons(z0, z1), Nil) -> c9 [1] MERGE(Nil, z0) -> c10 [1] SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) [1] SPLITMERGE(Nil, Cons(z01, Cons(z1'', z2')), Cons(z03, Cons(z11, z2''))) -> c12(MERGE(splitmerge(Cons(z01, Cons(z1'', z2')), Nil, Nil), splitmerge(Cons(z03, Cons(z11, z2'')), Nil, Nil)), MERGESORT(Cons(z01, Cons(z1'', z2')))) [1] SPLITMERGE(Nil, Cons(z01, Cons(z1'', z2')), Cons(z04, Nil)) -> c12(MERGE(splitmerge(Cons(z01, Cons(z1'', z2')), Nil, Nil), Cons(z04, Nil)), MERGESORT(Cons(z01, Cons(z1'', z2')))) [1] SPLITMERGE(Nil, Cons(z01, Cons(z1'', z2')), Nil) -> c12(MERGE(splitmerge(Cons(z01, Cons(z1'', z2')), Nil, Nil), Nil), MERGESORT(Cons(z01, Cons(z1'', z2')))) [1] SPLITMERGE(Nil, Cons(z01, Cons(z1'', z2')), z1) -> c12(MERGE(splitmerge(Cons(z01, Cons(z1'', z2')), Nil, Nil), Nil), MERGESORT(Cons(z01, Cons(z1'', z2')))) [1] SPLITMERGE(Nil, Cons(z02, Nil), Cons(z05, Cons(z12, z21))) -> c12(MERGE(Cons(z02, Nil), splitmerge(Cons(z05, Cons(z12, z21)), Nil, Nil)), MERGESORT(Cons(z02, Nil))) [1] SPLITMERGE(Nil, Cons(z02, Nil), Cons(z06, Nil)) -> c12(MERGE(Cons(z02, Nil), Cons(z06, Nil)), MERGESORT(Cons(z02, Nil))) [1] SPLITMERGE(Nil, Cons(z02, Nil), Nil) -> c12(MERGE(Cons(z02, Nil), Nil), MERGESORT(Cons(z02, Nil))) [1] SPLITMERGE(Nil, Cons(z02, Nil), z1) -> c12(MERGE(Cons(z02, Nil), Nil), MERGESORT(Cons(z02, Nil))) [1] SPLITMERGE(Nil, Nil, Cons(z07, Cons(z13, z22))) -> c12(MERGE(Nil, splitmerge(Cons(z07, Cons(z13, z22)), Nil, Nil)), MERGESORT(Nil)) [1] SPLITMERGE(Nil, Nil, Cons(z08, Nil)) -> c12(MERGE(Nil, Cons(z08, Nil)), MERGESORT(Nil)) [1] SPLITMERGE(Nil, Nil, Nil) -> c12(MERGE(Nil, Nil), MERGESORT(Nil)) [1] SPLITMERGE(Nil, Nil, z1) -> c12(MERGE(Nil, Nil), MERGESORT(Nil)) [1] SPLITMERGE(Nil, z0, Cons(z09, Cons(z14, z23))) -> c12(MERGE(Nil, splitmerge(Cons(z09, Cons(z14, z23)), Nil, Nil)), MERGESORT(z0)) [1] SPLITMERGE(Nil, z0, Cons(z010, Nil)) -> c12(MERGE(Nil, Cons(z010, Nil)), MERGESORT(z0)) [1] SPLITMERGE(Nil, z0, Nil) -> c12(MERGE(Nil, Nil), MERGESORT(z0)) [1] SPLITMERGE(Nil, z0, z1) -> c12(MERGE(Nil, Nil), MERGESORT(z0)) [1] SPLITMERGE(Nil, Cons(z011, Cons(z15, z24)), Cons(z013, Cons(z16, z25))) -> c13(MERGE(splitmerge(Cons(z011, Cons(z15, z24)), Nil, Nil), splitmerge(Cons(z013, Cons(z16, z25)), Nil, Nil)), MERGESORT(Cons(z013, Cons(z16, z25)))) [1] SPLITMERGE(Nil, Cons(z011, Cons(z15, z24)), Cons(z014, Nil)) -> c13(MERGE(splitmerge(Cons(z011, Cons(z15, z24)), Nil, Nil), Cons(z014, Nil)), MERGESORT(Cons(z014, Nil))) [1] SPLITMERGE(Nil, Cons(z011, Cons(z15, z24)), Nil) -> c13(MERGE(splitmerge(Cons(z011, Cons(z15, z24)), Nil, Nil), Nil), MERGESORT(Nil)) [1] SPLITMERGE(Nil, Cons(z011, Cons(z15, z24)), z1) -> c13(MERGE(splitmerge(Cons(z011, Cons(z15, z24)), Nil, Nil), Nil), MERGESORT(z1)) [1] SPLITMERGE(Nil, Cons(z012, Nil), Cons(z015, Cons(z17, z26))) -> c13(MERGE(Cons(z012, Nil), splitmerge(Cons(z015, Cons(z17, z26)), Nil, Nil)), MERGESORT(Cons(z015, Cons(z17, z26)))) [1] SPLITMERGE(Nil, Cons(z012, Nil), Cons(z016, Nil)) -> c13(MERGE(Cons(z012, Nil), Cons(z016, Nil)), MERGESORT(Cons(z016, Nil))) [1] SPLITMERGE(Nil, Cons(z012, Nil), Nil) -> c13(MERGE(Cons(z012, Nil), Nil), MERGESORT(Nil)) [1] SPLITMERGE(Nil, Cons(z012, Nil), z1) -> c13(MERGE(Cons(z012, Nil), Nil), MERGESORT(z1)) [1] SPLITMERGE(Nil, Nil, Cons(z017, Cons(z18, z27))) -> c13(MERGE(Nil, splitmerge(Cons(z017, Cons(z18, z27)), Nil, Nil)), MERGESORT(Cons(z017, Cons(z18, z27)))) [1] SPLITMERGE(Nil, Nil, Cons(z018, Nil)) -> c13(MERGE(Nil, Cons(z018, Nil)), MERGESORT(Cons(z018, Nil))) [1] SPLITMERGE(Nil, Nil, Nil) -> c13(MERGE(Nil, Nil), MERGESORT(Nil)) [1] SPLITMERGE(Nil, Nil, z1) -> c13(MERGE(Nil, Nil), MERGESORT(z1)) [1] SPLITMERGE(Nil, z0, Cons(z019, Cons(z19, z28))) -> c13(MERGE(Nil, splitmerge(Cons(z019, Cons(z19, z28)), Nil, Nil)), MERGESORT(Cons(z019, Cons(z19, z28)))) [1] SPLITMERGE(Nil, z0, Cons(z020, Nil)) -> c13(MERGE(Nil, Cons(z020, Nil)), MERGESORT(Cons(z020, Nil))) [1] SPLITMERGE(Nil, z0, Nil) -> c13(MERGE(Nil, Nil), MERGESORT(Nil)) [1] SPLITMERGE(Nil, z0, z1) -> c13(MERGE(Nil, Nil), MERGESORT(z1)) [1] <='(S(z0), S(z1)) -> c(<='(z0, z1)) [0] MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) [0] MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) [0] lteq(S(z0), S(z1)) -> lteq(z0, z1) [0] lteq(0, z0) -> True [0] lteq(S(z0), 0) -> False [0] mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) [0] mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) [0] mergesort(Nil) -> Nil [0] splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) [0] splitmerge(Nil, Cons(z021, Cons(z110, z29)), Cons(z023, Cons(z111, z210))) -> merge(splitmerge(Cons(z021, Cons(z110, z29)), Nil, Nil), splitmerge(Cons(z023, Cons(z111, z210)), Nil, Nil)) [0] splitmerge(Nil, Cons(z021, Cons(z110, z29)), Cons(z024, Nil)) -> merge(splitmerge(Cons(z021, Cons(z110, z29)), Nil, Nil), Cons(z024, Nil)) [0] splitmerge(Nil, Cons(z021, Cons(z110, z29)), Nil) -> merge(splitmerge(Cons(z021, Cons(z110, z29)), Nil, Nil), Nil) [0] splitmerge(Nil, Cons(z021, Cons(z110, z29)), z1) -> merge(splitmerge(Cons(z021, Cons(z110, z29)), Nil, Nil), Nil) [0] splitmerge(Nil, Cons(z022, Nil), Cons(z025, Cons(z112, z211))) -> merge(Cons(z022, Nil), splitmerge(Cons(z025, Cons(z112, z211)), Nil, Nil)) [0] splitmerge(Nil, Cons(z022, Nil), Cons(z026, Nil)) -> merge(Cons(z022, Nil), Cons(z026, Nil)) [0] splitmerge(Nil, Cons(z022, Nil), Nil) -> merge(Cons(z022, Nil), Nil) [0] splitmerge(Nil, Cons(z022, Nil), z1) -> merge(Cons(z022, Nil), Nil) [0] splitmerge(Nil, Nil, Cons(z027, Cons(z113, z212))) -> merge(Nil, splitmerge(Cons(z027, Cons(z113, z212)), Nil, Nil)) [0] splitmerge(Nil, Nil, Cons(z028, Nil)) -> merge(Nil, Cons(z028, Nil)) [0] splitmerge(Nil, Nil, Nil) -> merge(Nil, Nil) [0] splitmerge(Nil, Nil, z1) -> merge(Nil, Nil) [0] splitmerge(Nil, z0, Cons(z029, Cons(z114, z213))) -> merge(Nil, splitmerge(Cons(z029, Cons(z114, z213)), Nil, Nil)) [0] splitmerge(Nil, z0, Cons(z030, Nil)) -> merge(Nil, Cons(z030, Nil)) [0] splitmerge(Nil, z0, Nil) -> merge(Nil, Nil) [0] splitmerge(Nil, z0, z1) -> merge(Nil, Nil) [0] merge(Cons(S(z031), z1), Cons(S(z115), z3)) -> merge[Ite](lteq(z031, z115), Cons(S(z031), z1), Cons(S(z115), z3)) [0] merge(Cons(0, z1), Cons(z2, z3)) -> merge[Ite](True, Cons(0, z1), Cons(z2, z3)) [0] merge(Cons(S(z032), z1), Cons(0, z3)) -> merge[Ite](False, Cons(S(z032), z1), Cons(0, z3)) [0] merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](null_lteq, Cons(z0, z1), Cons(z2, z3)) [0] merge(Cons(z0, z1), Nil) -> Cons(z0, z1) [0] merge(Nil, z0) -> z0 [0] merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) [0] merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) [0] <='(v0, v1) -> const3 [0] MERGE[ITE](v0, v1, v2) -> const2 [0] lteq(v0, v1) -> null_lteq [0] mergesort(v0) -> Nil [0] splitmerge(v0, v1, v2) -> Nil [0] merge(v0, v1) -> Nil [0] merge[Ite](v0, v1, v2) -> Nil [0] The TRS has the following type information: MERGESORT :: Cons:Nil -> c5 Cons :: S:0 -> Cons:Nil -> Cons:Nil c5 :: c11:c12:c13 -> c5 SPLITMERGE :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c11:c12:c13 Nil :: Cons:Nil MERGE :: Cons:Nil -> Cons:Nil -> c8:c9:c10 c8 :: c3:c4:const2 -> c:const3 -> c8:c9:c10 MERGE[ITE] :: False:True:null_lteq -> Cons:Nil -> Cons:Nil -> c3:c4:const2 lteq :: S:0 -> S:0 -> False:True:null_lteq <=' :: S:0 -> S:0 -> c:const3 c9 :: c8:c9:c10 c10 :: c8:c9:c10 c11 :: c11:c12:c13 -> c11:c12:c13 c12 :: c8:c9:c10 -> c5 -> c11:c12:c13 mergesort :: Cons:Nil -> Cons:Nil c13 :: c8:c9:c10 -> c5 -> c11:c12:c13 S :: S:0 -> S:0 c :: c:const3 -> c:const3 False :: False:True:null_lteq c3 :: c8:c9:c10 -> c3:c4:const2 True :: False:True:null_lteq c4 :: c8:c9:c10 -> c3:c4:const2 0 :: S:0 splitmerge :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil merge :: Cons:Nil -> Cons:Nil -> Cons:Nil merge[Ite] :: False:True:null_lteq -> Cons:Nil -> Cons:Nil -> Cons:Nil const3 :: c:const3 const2 :: c3:c4:const2 null_lteq :: False:True:null_lteq const :: c5 const1 :: c11:c12:c13 Rewrite Strategy: INNERMOST ---------------------------------------- (37) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: Nil => 0 c9 => 1 c10 => 0 False => 1 True => 2 0 => 0 const3 => 0 const2 => 0 null_lteq => 0 const => 0 const1 => 0 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 <='(z, z') -{ 0 }-> 1 + <='(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z0 >= 0, z = 0, z' = z0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](lteq(z0', z1'), 1 + (1 + z0') + z1, 1 + (1 + z1') + z3) + <='(1 + z0', 1 + z1') :|: z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](2, 1 + 0 + z1, 1 + z2 + z3) + <='(0, z2) :|: z1 >= 0, z' = 1 + z2 + z3, z = 1 + 0 + z1, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](1, 1 + (1 + z0'') + z1, 1 + 0 + z3) + <='(1 + z0'', 0) :|: z1 >= 0, z = 1 + (1 + z0'') + z1, z' = 1 + 0 + z3, z0'' >= 0, z3 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](0, 1 + z0 + z1, 1 + z2 + z3) + <='(z0, z2) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z0, z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z1, z2) :|: z = 2, z'' = z2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z2 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z3, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), splitmerge(1 + z03 + (1 + z11 + z2''), 0, 0)) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z'' = z1, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 1 + z04 + 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 1 + z04 + 0, z04 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), splitmerge(1 + z013 + (1 + z16 + z25), 0, 0)) + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(z1) :|: z011 >= 0, z15 >= 0, z1 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0, z'' = z1 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(0) :|: z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 1 + z014 + 0) + MERGESORT(1 + z014 + 0) :|: z011 >= 0, z15 >= 0, z'' = 1 + z014 + 0, z' = 1 + z011 + (1 + z15 + z24), z014 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z017 + (1 + z18 + z27), 0, 0)) + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z019 + (1 + z19 + z28), 0, 0)) + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z0 >= 0, z = 0, z' = z0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z07 + (1 + z13 + z22), 0, 0)) + MERGESORT(0) :|: z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z09 + (1 + z14 + z23), 0, 0)) + MERGESORT(z0) :|: z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z0 >= 0, z09 >= 0, z = 0, z' = z0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z0) :|: z'' = 0, z0 >= 0, z = 0, z' = z0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z1) :|: z1 >= 0, z = 0, z' = 0, z'' = z1 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z1 >= 0, z = 0, z' = 0, z'' = z1 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z0 >= 0, z = 0, z' = z0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + z010 + 0) + MERGESORT(z0) :|: z'' = 1 + z010 + 0, z0 >= 0, z = 0, z' = z0, z010 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + z018 + 0) + MERGESORT(1 + z018 + 0) :|: z'' = 1 + z018 + 0, z = 0, z018 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + z020 + 0) + MERGESORT(1 + z020 + 0) :|: z'' = 1 + z020 + 0, z0 >= 0, z020 >= 0, z = 0, z' = z0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + z08 + 0) + MERGESORT(0) :|: z08 >= 0, z = 0, z'' = 1 + z08 + 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + z012 + 0, splitmerge(1 + z015 + (1 + z17 + z26), 0, 0)) + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' = 1 + z012 + 0, z012 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + z012 + 0, 0) + MERGESORT(z1) :|: z1 >= 0, z' = 1 + z012 + 0, z012 >= 0, z = 0, z'' = z1 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + z012 + 0, 0) + MERGESORT(0) :|: z'' = 0, z' = 1 + z012 + 0, z012 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + z012 + 0, 1 + z016 + 0) + MERGESORT(1 + z016 + 0) :|: z' = 1 + z012 + 0, z012 >= 0, z016 >= 0, z = 0, z'' = 1 + z016 + 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + z02 + 0, splitmerge(1 + z05 + (1 + z12 + z21), 0, 0)) + MERGESORT(1 + z02 + 0) :|: z21 >= 0, z' = 1 + z02 + 0, z02 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + z02 + 0, 0) + MERGESORT(1 + z02 + 0) :|: z'' = 0, z' = 1 + z02 + 0, z02 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + z02 + 0, 0) + MERGESORT(1 + z02 + 0) :|: z1 >= 0, z' = 1 + z02 + 0, z02 >= 0, z = 0, z'' = z1 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + z02 + 0, 1 + z06 + 0) + MERGESORT(1 + z02 + 0) :|: z'' = 1 + z06 + 0, z' = 1 + z02 + 0, z02 >= 0, z06 >= 0, z = 0 lteq(z, z') -{ 0 }-> lteq(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 lteq(z, z') -{ 0 }-> 2 :|: z0 >= 0, z = 0, z' = z0 lteq(z, z') -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 merge(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0 merge(z, z') -{ 0 }-> merge[Ite](lteq(z031, z115), 1 + (1 + z031) + z1, 1 + (1 + z115) + z3) :|: z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](2, 1 + 0 + z1, 1 + z2 + z3) :|: z1 >= 0, z' = 1 + z2 + z3, z = 1 + 0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](1, 1 + (1 + z032) + z1, 1 + 0 + z3) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' = 1 + 0 + z3, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](0, 1 + z0 + z1, 1 + z2 + z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + merge(z1, z2) :|: z = 2, z'' = z2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z2 >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + merge(z0, z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 mergesort(z) -{ 0 }-> 1 + z0 + 0 :|: z0 >= 0, z = 1 + z0 + 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z3, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), splitmerge(1 + z023 + (1 + z111 + z210), 0, 0)) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z021 >= 0, z110 >= 0, z1 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29), z'' = z1 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 1 + z024 + 0) :|: z021 >= 0, z110 >= 0, z29 >= 0, z024 >= 0, z'' = 1 + z024 + 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z027 + (1 + z113 + z212), 0, 0)) :|: z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z029 + (1 + z114 + z213), 0, 0)) :|: z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z0 >= 0, z = 0, z' = z0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z1 >= 0, z = 0, z' = 0, z'' = z1 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z0 >= 0, z = 0, z' = z0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + z028 + 0) :|: z'' = 1 + z028 + 0, z028 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + z030 + 0) :|: z0 >= 0, z = 0, z'' = 1 + z030 + 0, z' = z0, z030 >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + z022 + 0, splitmerge(1 + z025 + (1 + z112 + z211), 0, 0)) :|: z025 >= 0, z022 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0, z' = 1 + z022 + 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + z022 + 0, 0) :|: z'' = 0, z022 >= 0, z = 0, z' = 1 + z022 + 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + z022 + 0, 0) :|: z1 >= 0, z022 >= 0, z = 0, z' = 1 + z022 + 0, z'' = z1 splitmerge(z, z', z'') -{ 0 }-> merge(1 + z022 + 0, 1 + z026 + 0) :|: z022 >= 0, z'' = 1 + z026 + 0, z026 >= 0, z = 0, z' = 1 + z022 + 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 ---------------------------------------- (39) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + <='(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](lteq(z0', z1'), 1 + (1 + z0') + z1, 1 + (1 + z1') + z3) + <='(1 + z0', 1 + z1') :|: z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](2, 1 + 0 + (z - 1), 1 + z2 + z3) + <='(0, z2) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](1, 1 + (1 + z0'') + z1, 1 + 0 + (z' - 1)) + <='(1 + z0'', 0) :|: z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](0, 1 + z0 + z1, 1 + z2 + z3) + <='(z0, z2) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), splitmerge(1 + z03 + (1 + z11 + z2''), 0, 0)) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), splitmerge(1 + z013 + (1 + z16 + z25), 0, 0)) + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(z'') :|: z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(0) :|: z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z017 + (1 + z18 + z27), 0, 0)) + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z019 + (1 + z19 + z28), 0, 0)) + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z07 + (1 + z13 + z22), 0, 0)) + MERGESORT(0) :|: z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z09 + (1 + z14 + z23), 0, 0)) + MERGESORT(z') :|: z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(z') :|: z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(0) :|: z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z015 + (1 + z17 + z26), 0, 0)) + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z05 + (1 + z12 + z21), 0, 0)) + MERGESORT(1 + (z' - 1) + 0) :|: z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> merge[Ite](lteq(z031, z115), 1 + (1 + z031) + z1, 1 + (1 + z115) + z3) :|: z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + z2 + z3) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](1, 1 + (1 + z032) + z1, 1 + 0 + (z' - 1)) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> merge[Ite](0, 1 + z0 + z1, 1 + z2 + z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + merge(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + merge(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), splitmerge(1 + z023 + (1 + z111 + z210), 0, 0)) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 1 + (z'' - 1) + 0) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z027 + (1 + z113 + z212), 0, 0)) :|: z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z029 + (1 + z114 + z213), 0, 0)) :|: z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + z025 + (1 + z112 + z211), 0, 0)) :|: z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ---------------------------------------- (41) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { <=' } { lteq } { MERGE, MERGE[ITE] } { merge[Ite], merge } { splitmerge } { mergesort } { MERGESORT, SPLITMERGE } ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + <='(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](lteq(z0', z1'), 1 + (1 + z0') + z1, 1 + (1 + z1') + z3) + <='(1 + z0', 1 + z1') :|: z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](2, 1 + 0 + (z - 1), 1 + z2 + z3) + <='(0, z2) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](1, 1 + (1 + z0'') + z1, 1 + 0 + (z' - 1)) + <='(1 + z0'', 0) :|: z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](0, 1 + z0 + z1, 1 + z2 + z3) + <='(z0, z2) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), splitmerge(1 + z03 + (1 + z11 + z2''), 0, 0)) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), splitmerge(1 + z013 + (1 + z16 + z25), 0, 0)) + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(z'') :|: z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(0) :|: z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z017 + (1 + z18 + z27), 0, 0)) + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z019 + (1 + z19 + z28), 0, 0)) + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z07 + (1 + z13 + z22), 0, 0)) + MERGESORT(0) :|: z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z09 + (1 + z14 + z23), 0, 0)) + MERGESORT(z') :|: z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(z') :|: z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(0) :|: z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z015 + (1 + z17 + z26), 0, 0)) + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z05 + (1 + z12 + z21), 0, 0)) + MERGESORT(1 + (z' - 1) + 0) :|: z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> merge[Ite](lteq(z031, z115), 1 + (1 + z031) + z1, 1 + (1 + z115) + z3) :|: z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + z2 + z3) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](1, 1 + (1 + z032) + z1, 1 + 0 + (z' - 1)) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> merge[Ite](0, 1 + z0 + z1, 1 + z2 + z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + merge(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + merge(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), splitmerge(1 + z023 + (1 + z111 + z210), 0, 0)) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 1 + (z'' - 1) + 0) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z027 + (1 + z113 + z212), 0, 0)) :|: z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z029 + (1 + z114 + z213), 0, 0)) :|: z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + z025 + (1 + z112 + z211), 0, 0)) :|: z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {<='}, {lteq}, {MERGE,MERGE[ITE]}, {merge[Ite],merge}, {splitmerge}, {mergesort}, {MERGESORT,SPLITMERGE} ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + <='(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](lteq(z0', z1'), 1 + (1 + z0') + z1, 1 + (1 + z1') + z3) + <='(1 + z0', 1 + z1') :|: z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](2, 1 + 0 + (z - 1), 1 + z2 + z3) + <='(0, z2) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](1, 1 + (1 + z0'') + z1, 1 + 0 + (z' - 1)) + <='(1 + z0'', 0) :|: z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](0, 1 + z0 + z1, 1 + z2 + z3) + <='(z0, z2) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), splitmerge(1 + z03 + (1 + z11 + z2''), 0, 0)) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), splitmerge(1 + z013 + (1 + z16 + z25), 0, 0)) + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(z'') :|: z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(0) :|: z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z017 + (1 + z18 + z27), 0, 0)) + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z019 + (1 + z19 + z28), 0, 0)) + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z07 + (1 + z13 + z22), 0, 0)) + MERGESORT(0) :|: z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z09 + (1 + z14 + z23), 0, 0)) + MERGESORT(z') :|: z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(z') :|: z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(0) :|: z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z015 + (1 + z17 + z26), 0, 0)) + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z05 + (1 + z12 + z21), 0, 0)) + MERGESORT(1 + (z' - 1) + 0) :|: z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> merge[Ite](lteq(z031, z115), 1 + (1 + z031) + z1, 1 + (1 + z115) + z3) :|: z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + z2 + z3) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](1, 1 + (1 + z032) + z1, 1 + 0 + (z' - 1)) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> merge[Ite](0, 1 + z0 + z1, 1 + z2 + z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + merge(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + merge(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), splitmerge(1 + z023 + (1 + z111 + z210), 0, 0)) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 1 + (z'' - 1) + 0) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z027 + (1 + z113 + z212), 0, 0)) :|: z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z029 + (1 + z114 + z213), 0, 0)) :|: z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + z025 + (1 + z112 + z211), 0, 0)) :|: z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {<='}, {lteq}, {MERGE,MERGE[ITE]}, {merge[Ite],merge}, {splitmerge}, {mergesort}, {MERGESORT,SPLITMERGE} ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: <=' after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + <='(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](lteq(z0', z1'), 1 + (1 + z0') + z1, 1 + (1 + z1') + z3) + <='(1 + z0', 1 + z1') :|: z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](2, 1 + 0 + (z - 1), 1 + z2 + z3) + <='(0, z2) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](1, 1 + (1 + z0'') + z1, 1 + 0 + (z' - 1)) + <='(1 + z0'', 0) :|: z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](0, 1 + z0 + z1, 1 + z2 + z3) + <='(z0, z2) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), splitmerge(1 + z03 + (1 + z11 + z2''), 0, 0)) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), splitmerge(1 + z013 + (1 + z16 + z25), 0, 0)) + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(z'') :|: z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(0) :|: z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z017 + (1 + z18 + z27), 0, 0)) + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z019 + (1 + z19 + z28), 0, 0)) + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z07 + (1 + z13 + z22), 0, 0)) + MERGESORT(0) :|: z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z09 + (1 + z14 + z23), 0, 0)) + MERGESORT(z') :|: z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(z') :|: z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(0) :|: z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z015 + (1 + z17 + z26), 0, 0)) + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z05 + (1 + z12 + z21), 0, 0)) + MERGESORT(1 + (z' - 1) + 0) :|: z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> merge[Ite](lteq(z031, z115), 1 + (1 + z031) + z1, 1 + (1 + z115) + z3) :|: z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + z2 + z3) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](1, 1 + (1 + z032) + z1, 1 + 0 + (z' - 1)) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> merge[Ite](0, 1 + z0 + z1, 1 + z2 + z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + merge(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + merge(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), splitmerge(1 + z023 + (1 + z111 + z210), 0, 0)) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 1 + (z'' - 1) + 0) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z027 + (1 + z113 + z212), 0, 0)) :|: z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z029 + (1 + z114 + z213), 0, 0)) :|: z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + z025 + (1 + z112 + z211), 0, 0)) :|: z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {<='}, {lteq}, {MERGE,MERGE[ITE]}, {merge[Ite],merge}, {splitmerge}, {mergesort}, {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: ?, size: O(n^1) [z'] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: <=' after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + <='(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](lteq(z0', z1'), 1 + (1 + z0') + z1, 1 + (1 + z1') + z3) + <='(1 + z0', 1 + z1') :|: z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](2, 1 + 0 + (z - 1), 1 + z2 + z3) + <='(0, z2) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](1, 1 + (1 + z0'') + z1, 1 + 0 + (z' - 1)) + <='(1 + z0'', 0) :|: z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](0, 1 + z0 + z1, 1 + z2 + z3) + <='(z0, z2) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), splitmerge(1 + z03 + (1 + z11 + z2''), 0, 0)) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), splitmerge(1 + z013 + (1 + z16 + z25), 0, 0)) + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(z'') :|: z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(0) :|: z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z017 + (1 + z18 + z27), 0, 0)) + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z019 + (1 + z19 + z28), 0, 0)) + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z07 + (1 + z13 + z22), 0, 0)) + MERGESORT(0) :|: z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z09 + (1 + z14 + z23), 0, 0)) + MERGESORT(z') :|: z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(z') :|: z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(0) :|: z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z015 + (1 + z17 + z26), 0, 0)) + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z05 + (1 + z12 + z21), 0, 0)) + MERGESORT(1 + (z' - 1) + 0) :|: z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> merge[Ite](lteq(z031, z115), 1 + (1 + z031) + z1, 1 + (1 + z115) + z3) :|: z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + z2 + z3) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](1, 1 + (1 + z032) + z1, 1 + 0 + (z' - 1)) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> merge[Ite](0, 1 + z0 + z1, 1 + z2 + z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + merge(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + merge(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), splitmerge(1 + z023 + (1 + z111 + z210), 0, 0)) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 1 + (z'' - 1) + 0) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z027 + (1 + z113 + z212), 0, 0)) :|: z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z029 + (1 + z114 + z213), 0, 0)) :|: z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + z025 + (1 + z112 + z211), 0, 0)) :|: z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {lteq}, {MERGE,MERGE[ITE]}, {merge[Ite],merge}, {splitmerge}, {mergesort}, {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] ---------------------------------------- (49) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](lteq(z0', z1'), 1 + (1 + z0') + z1, 1 + (1 + z1') + z3) + s :|: s >= 0, s <= 1 + z1', z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](2, 1 + 0 + (z - 1), 1 + z2 + z3) + s' :|: s' >= 0, s' <= z2, z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](1, 1 + (1 + z0'') + z1, 1 + 0 + (z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](0, 1 + z0 + z1, 1 + z2 + z3) + s1 :|: s1 >= 0, s1 <= z2, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), splitmerge(1 + z03 + (1 + z11 + z2''), 0, 0)) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), splitmerge(1 + z013 + (1 + z16 + z25), 0, 0)) + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(z'') :|: z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(0) :|: z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z017 + (1 + z18 + z27), 0, 0)) + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z019 + (1 + z19 + z28), 0, 0)) + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z07 + (1 + z13 + z22), 0, 0)) + MERGESORT(0) :|: z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z09 + (1 + z14 + z23), 0, 0)) + MERGESORT(z') :|: z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(z') :|: z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(0) :|: z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z015 + (1 + z17 + z26), 0, 0)) + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z05 + (1 + z12 + z21), 0, 0)) + MERGESORT(1 + (z' - 1) + 0) :|: z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> merge[Ite](lteq(z031, z115), 1 + (1 + z031) + z1, 1 + (1 + z115) + z3) :|: z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + z2 + z3) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](1, 1 + (1 + z032) + z1, 1 + 0 + (z' - 1)) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> merge[Ite](0, 1 + z0 + z1, 1 + z2 + z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + merge(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + merge(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), splitmerge(1 + z023 + (1 + z111 + z210), 0, 0)) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 1 + (z'' - 1) + 0) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z027 + (1 + z113 + z212), 0, 0)) :|: z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z029 + (1 + z114 + z213), 0, 0)) :|: z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + z025 + (1 + z112 + z211), 0, 0)) :|: z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {lteq}, {MERGE,MERGE[ITE]}, {merge[Ite],merge}, {splitmerge}, {mergesort}, {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: lteq after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](lteq(z0', z1'), 1 + (1 + z0') + z1, 1 + (1 + z1') + z3) + s :|: s >= 0, s <= 1 + z1', z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](2, 1 + 0 + (z - 1), 1 + z2 + z3) + s' :|: s' >= 0, s' <= z2, z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](1, 1 + (1 + z0'') + z1, 1 + 0 + (z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](0, 1 + z0 + z1, 1 + z2 + z3) + s1 :|: s1 >= 0, s1 <= z2, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), splitmerge(1 + z03 + (1 + z11 + z2''), 0, 0)) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), splitmerge(1 + z013 + (1 + z16 + z25), 0, 0)) + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(z'') :|: z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(0) :|: z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z017 + (1 + z18 + z27), 0, 0)) + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z019 + (1 + z19 + z28), 0, 0)) + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z07 + (1 + z13 + z22), 0, 0)) + MERGESORT(0) :|: z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z09 + (1 + z14 + z23), 0, 0)) + MERGESORT(z') :|: z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(z') :|: z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(0) :|: z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z015 + (1 + z17 + z26), 0, 0)) + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z05 + (1 + z12 + z21), 0, 0)) + MERGESORT(1 + (z' - 1) + 0) :|: z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> merge[Ite](lteq(z031, z115), 1 + (1 + z031) + z1, 1 + (1 + z115) + z3) :|: z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + z2 + z3) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](1, 1 + (1 + z032) + z1, 1 + 0 + (z' - 1)) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> merge[Ite](0, 1 + z0 + z1, 1 + z2 + z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + merge(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + merge(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), splitmerge(1 + z023 + (1 + z111 + z210), 0, 0)) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 1 + (z'' - 1) + 0) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z027 + (1 + z113 + z212), 0, 0)) :|: z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z029 + (1 + z114 + z213), 0, 0)) :|: z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + z025 + (1 + z112 + z211), 0, 0)) :|: z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {lteq}, {MERGE,MERGE[ITE]}, {merge[Ite],merge}, {splitmerge}, {mergesort}, {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] lteq: runtime: ?, size: O(1) [2] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: lteq after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](lteq(z0', z1'), 1 + (1 + z0') + z1, 1 + (1 + z1') + z3) + s :|: s >= 0, s <= 1 + z1', z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](2, 1 + 0 + (z - 1), 1 + z2 + z3) + s' :|: s' >= 0, s' <= z2, z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](1, 1 + (1 + z0'') + z1, 1 + 0 + (z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](0, 1 + z0 + z1, 1 + z2 + z3) + s1 :|: s1 >= 0, s1 <= z2, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), splitmerge(1 + z03 + (1 + z11 + z2''), 0, 0)) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), splitmerge(1 + z013 + (1 + z16 + z25), 0, 0)) + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(z'') :|: z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(0) :|: z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z017 + (1 + z18 + z27), 0, 0)) + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z019 + (1 + z19 + z28), 0, 0)) + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z07 + (1 + z13 + z22), 0, 0)) + MERGESORT(0) :|: z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z09 + (1 + z14 + z23), 0, 0)) + MERGESORT(z') :|: z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(z') :|: z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(0) :|: z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z015 + (1 + z17 + z26), 0, 0)) + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z05 + (1 + z12 + z21), 0, 0)) + MERGESORT(1 + (z' - 1) + 0) :|: z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> merge[Ite](lteq(z031, z115), 1 + (1 + z031) + z1, 1 + (1 + z115) + z3) :|: z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + z2 + z3) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](1, 1 + (1 + z032) + z1, 1 + 0 + (z' - 1)) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> merge[Ite](0, 1 + z0 + z1, 1 + z2 + z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + merge(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + merge(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), splitmerge(1 + z023 + (1 + z111 + z210), 0, 0)) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 1 + (z'' - 1) + 0) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z027 + (1 + z113 + z212), 0, 0)) :|: z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z029 + (1 + z114 + z213), 0, 0)) :|: z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + z025 + (1 + z112 + z211), 0, 0)) :|: z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {MERGE,MERGE[ITE]}, {merge[Ite],merge}, {splitmerge}, {mergesort}, {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] lteq: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (55) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](s3, 1 + (1 + z0') + z1, 1 + (1 + z1') + z3) + s :|: s3 >= 0, s3 <= 2, s >= 0, s <= 1 + z1', z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](2, 1 + 0 + (z - 1), 1 + z2 + z3) + s' :|: s' >= 0, s' <= z2, z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](1, 1 + (1 + z0'') + z1, 1 + 0 + (z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](0, 1 + z0 + z1, 1 + z2 + z3) + s1 :|: s1 >= 0, s1 <= z2, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), splitmerge(1 + z03 + (1 + z11 + z2''), 0, 0)) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), splitmerge(1 + z013 + (1 + z16 + z25), 0, 0)) + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(z'') :|: z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(0) :|: z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z017 + (1 + z18 + z27), 0, 0)) + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z019 + (1 + z19 + z28), 0, 0)) + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z07 + (1 + z13 + z22), 0, 0)) + MERGESORT(0) :|: z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z09 + (1 + z14 + z23), 0, 0)) + MERGESORT(z') :|: z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(z') :|: z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(0) :|: z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z015 + (1 + z17 + z26), 0, 0)) + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z05 + (1 + z12 + z21), 0, 0)) + MERGESORT(1 + (z' - 1) + 0) :|: z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 lteq(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> merge[Ite](s5, 1 + (1 + z031) + z1, 1 + (1 + z115) + z3) :|: s5 >= 0, s5 <= 2, z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + z2 + z3) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](1, 1 + (1 + z032) + z1, 1 + 0 + (z' - 1)) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> merge[Ite](0, 1 + z0 + z1, 1 + z2 + z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + merge(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + merge(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), splitmerge(1 + z023 + (1 + z111 + z210), 0, 0)) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 1 + (z'' - 1) + 0) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z027 + (1 + z113 + z212), 0, 0)) :|: z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z029 + (1 + z114 + z213), 0, 0)) :|: z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + z025 + (1 + z112 + z211), 0, 0)) :|: z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {MERGE,MERGE[ITE]}, {merge[Ite],merge}, {splitmerge}, {mergesort}, {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] lteq: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: MERGE after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 4 + 2*z + z*z' + 7*z' + 2*z'^2 Computed SIZE bound using KoAT for: MERGE[ITE] after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 10 + 4*z' + 2*z'*z'' + 14*z'' + 4*z''^2 ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](s3, 1 + (1 + z0') + z1, 1 + (1 + z1') + z3) + s :|: s3 >= 0, s3 <= 2, s >= 0, s <= 1 + z1', z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](2, 1 + 0 + (z - 1), 1 + z2 + z3) + s' :|: s' >= 0, s' <= z2, z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](1, 1 + (1 + z0'') + z1, 1 + 0 + (z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](0, 1 + z0 + z1, 1 + z2 + z3) + s1 :|: s1 >= 0, s1 <= z2, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), splitmerge(1 + z03 + (1 + z11 + z2''), 0, 0)) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), splitmerge(1 + z013 + (1 + z16 + z25), 0, 0)) + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(z'') :|: z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(0) :|: z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z017 + (1 + z18 + z27), 0, 0)) + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z019 + (1 + z19 + z28), 0, 0)) + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z07 + (1 + z13 + z22), 0, 0)) + MERGESORT(0) :|: z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z09 + (1 + z14 + z23), 0, 0)) + MERGESORT(z') :|: z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(z') :|: z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(0) :|: z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z015 + (1 + z17 + z26), 0, 0)) + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z05 + (1 + z12 + z21), 0, 0)) + MERGESORT(1 + (z' - 1) + 0) :|: z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 lteq(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> merge[Ite](s5, 1 + (1 + z031) + z1, 1 + (1 + z115) + z3) :|: s5 >= 0, s5 <= 2, z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + z2 + z3) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](1, 1 + (1 + z032) + z1, 1 + 0 + (z' - 1)) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> merge[Ite](0, 1 + z0 + z1, 1 + z2 + z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + merge(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + merge(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), splitmerge(1 + z023 + (1 + z111 + z210), 0, 0)) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 1 + (z'' - 1) + 0) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z027 + (1 + z113 + z212), 0, 0)) :|: z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z029 + (1 + z114 + z213), 0, 0)) :|: z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + z025 + (1 + z112 + z211), 0, 0)) :|: z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {MERGE,MERGE[ITE]}, {merge[Ite],merge}, {splitmerge}, {mergesort}, {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] lteq: runtime: O(1) [0], size: O(1) [2] MERGE: runtime: ?, size: O(n^2) [4 + 2*z + z*z' + 7*z' + 2*z'^2] MERGE[ITE]: runtime: ?, size: O(n^2) [10 + 4*z' + 2*z'*z'' + 14*z'' + 4*z''^2] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: MERGE after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + z + 2*z' Computed RUNTIME bound using CoFloCo for: MERGE[ITE] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' + 2*z'' ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](s3, 1 + (1 + z0') + z1, 1 + (1 + z1') + z3) + s :|: s3 >= 0, s3 <= 2, s >= 0, s <= 1 + z1', z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](2, 1 + 0 + (z - 1), 1 + z2 + z3) + s' :|: s' >= 0, s' <= z2, z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](1, 1 + (1 + z0'') + z1, 1 + 0 + (z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](0, 1 + z0 + z1, 1 + z2 + z3) + s1 :|: s1 >= 0, s1 <= z2, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), splitmerge(1 + z03 + (1 + z11 + z2''), 0, 0)) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), splitmerge(1 + z013 + (1 + z16 + z25), 0, 0)) + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(z'') :|: z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(0) :|: z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z017 + (1 + z18 + z27), 0, 0)) + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z019 + (1 + z19 + z28), 0, 0)) + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z07 + (1 + z13 + z22), 0, 0)) + MERGESORT(0) :|: z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z09 + (1 + z14 + z23), 0, 0)) + MERGESORT(z') :|: z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(z'') :|: z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 0) + MERGESORT(0) :|: z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(z') :|: z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(0) :|: z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z015 + (1 + z17 + z26), 0, 0)) + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z05 + (1 + z12 + z21), 0, 0)) + MERGESORT(1 + (z' - 1) + 0) :|: z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 0) + MERGESORT(1 + (z' - 1) + 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 lteq(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> merge[Ite](s5, 1 + (1 + z031) + z1, 1 + (1 + z115) + z3) :|: s5 >= 0, s5 <= 2, z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + z2 + z3) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](1, 1 + (1 + z032) + z1, 1 + 0 + (z' - 1)) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> merge[Ite](0, 1 + z0 + z1, 1 + z2 + z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + merge(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + merge(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), splitmerge(1 + z023 + (1 + z111 + z210), 0, 0)) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 1 + (z'' - 1) + 0) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z027 + (1 + z113 + z212), 0, 0)) :|: z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z029 + (1 + z114 + z213), 0, 0)) :|: z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + z025 + (1 + z112 + z211), 0, 0)) :|: z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {merge[Ite],merge}, {splitmerge}, {mergesort}, {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] lteq: runtime: O(1) [0], size: O(1) [2] MERGE: runtime: O(n^1) [3 + z + 2*z'], size: O(n^2) [4 + 2*z + z*z' + 7*z' + 2*z'^2] MERGE[ITE]: runtime: O(n^1) [2 + z' + 2*z''], size: O(n^2) [10 + 4*z' + 2*z'*z'' + 14*z'' + 4*z''^2] ---------------------------------------- (61) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 9 + z0' + z1 + 2*z1' + 2*z3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 14 * (1 + (1 + z1') + z3) + 2 * ((1 + (1 + z1') + z3) * (1 + (1 + z0') + z1)) + 4 * ((1 + (1 + z1') + z3) * (1 + (1 + z1') + z3)) + 4 * (1 + (1 + z0') + z1) + 10, s3 >= 0, s3 <= 2, s >= 0, s <= 1 + z1', z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 5 + z + 2*z2 + 2*z3 }-> 1 + s7 + s' :|: s7 >= 0, s7 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + 0 + (z - 1))) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + 0 + (z - 1)) + 10, s' >= 0, s' <= z2, z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 5 + 2*z' + z0'' + z1 }-> 1 + s8 + s'' :|: s8 >= 0, s8 <= 14 * (1 + 0 + (z' - 1)) + 2 * ((1 + 0 + (z' - 1)) * (1 + (1 + z0'') + z1)) + 4 * ((1 + 0 + (z' - 1)) * (1 + 0 + (z' - 1))) + 4 * (1 + (1 + z0'') + z1) + 10, s'' >= 0, s'' <= 0, z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 6 + z0 + z1 + 2*z2 + 2*z3 }-> 1 + s9 + s1 :|: s9 >= 0, s9 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + z0 + z1)) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + z0 + z1) + 10, s1 >= 0, s1 <= z2, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 3 + z' + 2*z2 }-> 1 + s27 :|: s27 >= 0, s27 <= 4 + 2 * z' + 7 * z2 + z2 * z' + 2 * (z2 * z2), z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 3 + 2*z'' + z1 }-> 1 + s28 :|: s28 >= 0, s28 <= 4 + 2 * z1 + 7 * z'' + z'' * z1 + 2 * (z'' * z''), z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s10 + MERGESORT(1 + (z' - 1) + 0) :|: s10 >= 0, s10 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s11 + MERGESORT(1 + (z' - 1) + 0) :|: s11 >= 0, s11 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s12 + MERGESORT(1 + (z' - 1) + 0) :|: s12 >= 0, s12 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s13 + MERGESORT(0) :|: s13 >= 0, s13 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s14 + MERGESORT(0) :|: s14 >= 0, s14 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s15 + MERGESORT(0) :|: s15 >= 0, s15 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s16 + MERGESORT(z') :|: s16 >= 0, s16 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s17 + MERGESORT(z') :|: s17 >= 0, s17 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s18 + MERGESORT(z') :|: s18 >= 0, s18 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s19 + MERGESORT(1 + (z'' - 1) + 0) :|: s19 >= 0, s19 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s20 + MERGESORT(0) :|: s20 >= 0, s20 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s21 + MERGESORT(z'') :|: s21 >= 0, s21 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s22 + MERGESORT(1 + (z'' - 1) + 0) :|: s22 >= 0, s22 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s23 + MERGESORT(z'') :|: s23 >= 0, s23 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s24 + MERGESORT(1 + (z'' - 1) + 0) :|: s24 >= 0, s24 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s25 + MERGESORT(0) :|: s25 >= 0, s25 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s26 + MERGESORT(z'') :|: s26 >= 0, s26 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), splitmerge(1 + z03 + (1 + z11 + z2''), 0, 0)) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), splitmerge(1 + z013 + (1 + z16 + z25), 0, 0)) + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(z'') :|: z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(0) :|: z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z017 + (1 + z18 + z27), 0, 0)) + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z019 + (1 + z19 + z28), 0, 0)) + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z07 + (1 + z13 + z22), 0, 0)) + MERGESORT(0) :|: z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z09 + (1 + z14 + z23), 0, 0)) + MERGESORT(z') :|: z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z015 + (1 + z17 + z26), 0, 0)) + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z05 + (1 + z12 + z21), 0, 0)) + MERGESORT(1 + (z' - 1) + 0) :|: z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 lteq(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> merge[Ite](s5, 1 + (1 + z031) + z1, 1 + (1 + z115) + z3) :|: s5 >= 0, s5 <= 2, z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + z2 + z3) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](1, 1 + (1 + z032) + z1, 1 + 0 + (z' - 1)) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> merge[Ite](0, 1 + z0 + z1, 1 + z2 + z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + merge(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + merge(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), splitmerge(1 + z023 + (1 + z111 + z210), 0, 0)) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 1 + (z'' - 1) + 0) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z027 + (1 + z113 + z212), 0, 0)) :|: z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z029 + (1 + z114 + z213), 0, 0)) :|: z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + z025 + (1 + z112 + z211), 0, 0)) :|: z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {merge[Ite],merge}, {splitmerge}, {mergesort}, {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] lteq: runtime: O(1) [0], size: O(1) [2] MERGE: runtime: O(n^1) [3 + z + 2*z'], size: O(n^2) [4 + 2*z + z*z' + 7*z' + 2*z'^2] MERGE[ITE]: runtime: O(n^1) [2 + z' + 2*z''], size: O(n^2) [10 + 4*z' + 2*z'*z'' + 14*z'' + 4*z''^2] ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: merge[Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' + z'' Computed SIZE bound using CoFloCo for: merge after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 9 + z0' + z1 + 2*z1' + 2*z3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 14 * (1 + (1 + z1') + z3) + 2 * ((1 + (1 + z1') + z3) * (1 + (1 + z0') + z1)) + 4 * ((1 + (1 + z1') + z3) * (1 + (1 + z1') + z3)) + 4 * (1 + (1 + z0') + z1) + 10, s3 >= 0, s3 <= 2, s >= 0, s <= 1 + z1', z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 5 + z + 2*z2 + 2*z3 }-> 1 + s7 + s' :|: s7 >= 0, s7 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + 0 + (z - 1))) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + 0 + (z - 1)) + 10, s' >= 0, s' <= z2, z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 5 + 2*z' + z0'' + z1 }-> 1 + s8 + s'' :|: s8 >= 0, s8 <= 14 * (1 + 0 + (z' - 1)) + 2 * ((1 + 0 + (z' - 1)) * (1 + (1 + z0'') + z1)) + 4 * ((1 + 0 + (z' - 1)) * (1 + 0 + (z' - 1))) + 4 * (1 + (1 + z0'') + z1) + 10, s'' >= 0, s'' <= 0, z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 6 + z0 + z1 + 2*z2 + 2*z3 }-> 1 + s9 + s1 :|: s9 >= 0, s9 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + z0 + z1)) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + z0 + z1) + 10, s1 >= 0, s1 <= z2, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 3 + z' + 2*z2 }-> 1 + s27 :|: s27 >= 0, s27 <= 4 + 2 * z' + 7 * z2 + z2 * z' + 2 * (z2 * z2), z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 3 + 2*z'' + z1 }-> 1 + s28 :|: s28 >= 0, s28 <= 4 + 2 * z1 + 7 * z'' + z'' * z1 + 2 * (z'' * z''), z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s10 + MERGESORT(1 + (z' - 1) + 0) :|: s10 >= 0, s10 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s11 + MERGESORT(1 + (z' - 1) + 0) :|: s11 >= 0, s11 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s12 + MERGESORT(1 + (z' - 1) + 0) :|: s12 >= 0, s12 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s13 + MERGESORT(0) :|: s13 >= 0, s13 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s14 + MERGESORT(0) :|: s14 >= 0, s14 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s15 + MERGESORT(0) :|: s15 >= 0, s15 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s16 + MERGESORT(z') :|: s16 >= 0, s16 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s17 + MERGESORT(z') :|: s17 >= 0, s17 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s18 + MERGESORT(z') :|: s18 >= 0, s18 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s19 + MERGESORT(1 + (z'' - 1) + 0) :|: s19 >= 0, s19 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s20 + MERGESORT(0) :|: s20 >= 0, s20 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s21 + MERGESORT(z'') :|: s21 >= 0, s21 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s22 + MERGESORT(1 + (z'' - 1) + 0) :|: s22 >= 0, s22 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s23 + MERGESORT(z'') :|: s23 >= 0, s23 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s24 + MERGESORT(1 + (z'' - 1) + 0) :|: s24 >= 0, s24 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s25 + MERGESORT(0) :|: s25 >= 0, s25 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s26 + MERGESORT(z'') :|: s26 >= 0, s26 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), splitmerge(1 + z03 + (1 + z11 + z2''), 0, 0)) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), splitmerge(1 + z013 + (1 + z16 + z25), 0, 0)) + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(z'') :|: z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(0) :|: z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z017 + (1 + z18 + z27), 0, 0)) + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z019 + (1 + z19 + z28), 0, 0)) + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z07 + (1 + z13 + z22), 0, 0)) + MERGESORT(0) :|: z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z09 + (1 + z14 + z23), 0, 0)) + MERGESORT(z') :|: z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z015 + (1 + z17 + z26), 0, 0)) + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z05 + (1 + z12 + z21), 0, 0)) + MERGESORT(1 + (z' - 1) + 0) :|: z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 lteq(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> merge[Ite](s5, 1 + (1 + z031) + z1, 1 + (1 + z115) + z3) :|: s5 >= 0, s5 <= 2, z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + z2 + z3) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](1, 1 + (1 + z032) + z1, 1 + 0 + (z' - 1)) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> merge[Ite](0, 1 + z0 + z1, 1 + z2 + z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + merge(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + merge(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), splitmerge(1 + z023 + (1 + z111 + z210), 0, 0)) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 1 + (z'' - 1) + 0) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z027 + (1 + z113 + z212), 0, 0)) :|: z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z029 + (1 + z114 + z213), 0, 0)) :|: z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + z025 + (1 + z112 + z211), 0, 0)) :|: z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {merge[Ite],merge}, {splitmerge}, {mergesort}, {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] lteq: runtime: O(1) [0], size: O(1) [2] MERGE: runtime: O(n^1) [3 + z + 2*z'], size: O(n^2) [4 + 2*z + z*z' + 7*z' + 2*z'^2] MERGE[ITE]: runtime: O(n^1) [2 + z' + 2*z''], size: O(n^2) [10 + 4*z' + 2*z'*z'' + 14*z'' + 4*z''^2] merge[Ite]: runtime: ?, size: O(n^1) [z' + z''] merge: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: merge[Ite] after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed RUNTIME bound using CoFloCo for: merge after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 9 + z0' + z1 + 2*z1' + 2*z3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 14 * (1 + (1 + z1') + z3) + 2 * ((1 + (1 + z1') + z3) * (1 + (1 + z0') + z1)) + 4 * ((1 + (1 + z1') + z3) * (1 + (1 + z1') + z3)) + 4 * (1 + (1 + z0') + z1) + 10, s3 >= 0, s3 <= 2, s >= 0, s <= 1 + z1', z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 5 + z + 2*z2 + 2*z3 }-> 1 + s7 + s' :|: s7 >= 0, s7 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + 0 + (z - 1))) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + 0 + (z - 1)) + 10, s' >= 0, s' <= z2, z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 5 + 2*z' + z0'' + z1 }-> 1 + s8 + s'' :|: s8 >= 0, s8 <= 14 * (1 + 0 + (z' - 1)) + 2 * ((1 + 0 + (z' - 1)) * (1 + (1 + z0'') + z1)) + 4 * ((1 + 0 + (z' - 1)) * (1 + 0 + (z' - 1))) + 4 * (1 + (1 + z0'') + z1) + 10, s'' >= 0, s'' <= 0, z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 6 + z0 + z1 + 2*z2 + 2*z3 }-> 1 + s9 + s1 :|: s9 >= 0, s9 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + z0 + z1)) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + z0 + z1) + 10, s1 >= 0, s1 <= z2, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 3 + z' + 2*z2 }-> 1 + s27 :|: s27 >= 0, s27 <= 4 + 2 * z' + 7 * z2 + z2 * z' + 2 * (z2 * z2), z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 3 + 2*z'' + z1 }-> 1 + s28 :|: s28 >= 0, s28 <= 4 + 2 * z1 + 7 * z'' + z'' * z1 + 2 * (z'' * z''), z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s10 + MERGESORT(1 + (z' - 1) + 0) :|: s10 >= 0, s10 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s11 + MERGESORT(1 + (z' - 1) + 0) :|: s11 >= 0, s11 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s12 + MERGESORT(1 + (z' - 1) + 0) :|: s12 >= 0, s12 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s13 + MERGESORT(0) :|: s13 >= 0, s13 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s14 + MERGESORT(0) :|: s14 >= 0, s14 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s15 + MERGESORT(0) :|: s15 >= 0, s15 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s16 + MERGESORT(z') :|: s16 >= 0, s16 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s17 + MERGESORT(z') :|: s17 >= 0, s17 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s18 + MERGESORT(z') :|: s18 >= 0, s18 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s19 + MERGESORT(1 + (z'' - 1) + 0) :|: s19 >= 0, s19 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s20 + MERGESORT(0) :|: s20 >= 0, s20 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s21 + MERGESORT(z'') :|: s21 >= 0, s21 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s22 + MERGESORT(1 + (z'' - 1) + 0) :|: s22 >= 0, s22 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s23 + MERGESORT(z'') :|: s23 >= 0, s23 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s24 + MERGESORT(1 + (z'' - 1) + 0) :|: s24 >= 0, s24 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s25 + MERGESORT(0) :|: s25 >= 0, s25 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s26 + MERGESORT(z'') :|: s26 >= 0, s26 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), splitmerge(1 + z03 + (1 + z11 + z2''), 0, 0)) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), splitmerge(1 + z013 + (1 + z16 + z25), 0, 0)) + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(z'') :|: z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(0) :|: z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z017 + (1 + z18 + z27), 0, 0)) + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z019 + (1 + z19 + z28), 0, 0)) + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z07 + (1 + z13 + z22), 0, 0)) + MERGESORT(0) :|: z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z09 + (1 + z14 + z23), 0, 0)) + MERGESORT(z') :|: z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z015 + (1 + z17 + z26), 0, 0)) + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z05 + (1 + z12 + z21), 0, 0)) + MERGESORT(1 + (z' - 1) + 0) :|: z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 lteq(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> merge[Ite](s5, 1 + (1 + z031) + z1, 1 + (1 + z115) + z3) :|: s5 >= 0, s5 <= 2, z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + z2 + z3) :|: z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> merge[Ite](1, 1 + (1 + z032) + z1, 1 + 0 + (z' - 1)) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> merge[Ite](0, 1 + z0 + z1, 1 + z2 + z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + merge(z1, z'') :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + merge(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), splitmerge(1 + z023 + (1 + z111 + z210), 0, 0)) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 1 + (z'' - 1) + 0) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z027 + (1 + z113 + z212), 0, 0)) :|: z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z029 + (1 + z114 + z213), 0, 0)) :|: z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 0) :|: z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, 1 + (z'' - 1) + 0) :|: z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + z025 + (1 + z112 + z211), 0, 0)) :|: z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {splitmerge}, {mergesort}, {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] lteq: runtime: O(1) [0], size: O(1) [2] MERGE: runtime: O(n^1) [3 + z + 2*z'], size: O(n^2) [4 + 2*z + z*z' + 7*z' + 2*z'^2] MERGE[ITE]: runtime: O(n^1) [2 + z' + 2*z''], size: O(n^2) [10 + 4*z' + 2*z'*z'' + 14*z'' + 4*z''^2] merge[Ite]: runtime: O(1) [0], size: O(n^1) [z' + z''] merge: runtime: O(1) [0], size: O(n^1) [z + z'] ---------------------------------------- (67) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 9 + z0' + z1 + 2*z1' + 2*z3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 14 * (1 + (1 + z1') + z3) + 2 * ((1 + (1 + z1') + z3) * (1 + (1 + z0') + z1)) + 4 * ((1 + (1 + z1') + z3) * (1 + (1 + z1') + z3)) + 4 * (1 + (1 + z0') + z1) + 10, s3 >= 0, s3 <= 2, s >= 0, s <= 1 + z1', z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 5 + z + 2*z2 + 2*z3 }-> 1 + s7 + s' :|: s7 >= 0, s7 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + 0 + (z - 1))) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + 0 + (z - 1)) + 10, s' >= 0, s' <= z2, z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 5 + 2*z' + z0'' + z1 }-> 1 + s8 + s'' :|: s8 >= 0, s8 <= 14 * (1 + 0 + (z' - 1)) + 2 * ((1 + 0 + (z' - 1)) * (1 + (1 + z0'') + z1)) + 4 * ((1 + 0 + (z' - 1)) * (1 + 0 + (z' - 1))) + 4 * (1 + (1 + z0'') + z1) + 10, s'' >= 0, s'' <= 0, z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 6 + z0 + z1 + 2*z2 + 2*z3 }-> 1 + s9 + s1 :|: s9 >= 0, s9 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + z0 + z1)) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + z0 + z1) + 10, s1 >= 0, s1 <= z2, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 3 + z' + 2*z2 }-> 1 + s27 :|: s27 >= 0, s27 <= 4 + 2 * z' + 7 * z2 + z2 * z' + 2 * (z2 * z2), z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 3 + 2*z'' + z1 }-> 1 + s28 :|: s28 >= 0, s28 <= 4 + 2 * z1 + 7 * z'' + z'' * z1 + 2 * (z'' * z''), z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s10 + MERGESORT(1 + (z' - 1) + 0) :|: s10 >= 0, s10 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s11 + MERGESORT(1 + (z' - 1) + 0) :|: s11 >= 0, s11 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s12 + MERGESORT(1 + (z' - 1) + 0) :|: s12 >= 0, s12 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s13 + MERGESORT(0) :|: s13 >= 0, s13 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s14 + MERGESORT(0) :|: s14 >= 0, s14 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s15 + MERGESORT(0) :|: s15 >= 0, s15 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s16 + MERGESORT(z') :|: s16 >= 0, s16 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s17 + MERGESORT(z') :|: s17 >= 0, s17 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s18 + MERGESORT(z') :|: s18 >= 0, s18 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s19 + MERGESORT(1 + (z'' - 1) + 0) :|: s19 >= 0, s19 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s20 + MERGESORT(0) :|: s20 >= 0, s20 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s21 + MERGESORT(z'') :|: s21 >= 0, s21 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s22 + MERGESORT(1 + (z'' - 1) + 0) :|: s22 >= 0, s22 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s23 + MERGESORT(z'') :|: s23 >= 0, s23 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s24 + MERGESORT(1 + (z'' - 1) + 0) :|: s24 >= 0, s24 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s25 + MERGESORT(0) :|: s25 >= 0, s25 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s26 + MERGESORT(z'') :|: s26 >= 0, s26 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), splitmerge(1 + z03 + (1 + z11 + z2''), 0, 0)) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), splitmerge(1 + z013 + (1 + z16 + z25), 0, 0)) + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(z'') :|: z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(0) :|: z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z017 + (1 + z18 + z27), 0, 0)) + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z019 + (1 + z19 + z28), 0, 0)) + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z07 + (1 + z13 + z22), 0, 0)) + MERGESORT(0) :|: z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z09 + (1 + z14 + z23), 0, 0)) + MERGESORT(z') :|: z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z015 + (1 + z17 + z26), 0, 0)) + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z05 + (1 + z12 + z21), 0, 0)) + MERGESORT(1 + (z' - 1) + 0) :|: z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 lteq(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> s38 :|: s38 >= 0, s38 <= 1 + (1 + z031) + z1 + (1 + (1 + z115) + z3), s5 >= 0, s5 <= 2, z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> s39 :|: s39 >= 0, s39 <= 1 + 0 + (z - 1) + (1 + z2 + z3), z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> s40 :|: s40 >= 0, s40 <= 1 + (1 + z032) + z1 + (1 + 0 + (z' - 1)), z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> s41 :|: s41 >= 0, s41 <= 1 + z0 + z1 + (1 + z2 + z3), z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + s43 :|: s43 >= 0, s43 <= z1 + z'', z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + s42 :|: s42 >= 0, s42 <= z' + z2, z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> s29 :|: s29 >= 0, s29 <= 1 + (z' - 1) + 0 + (1 + (z'' - 1) + 0), z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s30 :|: s30 >= 0, s30 <= 1 + (z' - 1) + 0 + 0, z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s31 :|: s31 >= 0, s31 <= 1 + (z' - 1) + 0 + 0, z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 0 + (1 + (z'' - 1) + 0), z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 0 + 0, z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= 0 + 0, z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 0 + (1 + (z'' - 1) + 0), z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> s36 :|: s36 >= 0, s36 <= 0 + 0, z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s37 :|: s37 >= 0, s37 <= 0 + 0, z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), splitmerge(1 + z023 + (1 + z111 + z210), 0, 0)) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 1 + (z'' - 1) + 0) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z027 + (1 + z113 + z212), 0, 0)) :|: z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z029 + (1 + z114 + z213), 0, 0)) :|: z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + z025 + (1 + z112 + z211), 0, 0)) :|: z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {splitmerge}, {mergesort}, {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] lteq: runtime: O(1) [0], size: O(1) [2] MERGE: runtime: O(n^1) [3 + z + 2*z'], size: O(n^2) [4 + 2*z + z*z' + 7*z' + 2*z'^2] MERGE[ITE]: runtime: O(n^1) [2 + z' + 2*z''], size: O(n^2) [10 + 4*z' + 2*z'*z'' + 14*z'' + 4*z''^2] merge[Ite]: runtime: O(1) [0], size: O(n^1) [z' + z''] merge: runtime: O(1) [0], size: O(n^1) [z + z'] ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: splitmerge after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 14 + 29*z + 29*z' + 29*z'' ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 9 + z0' + z1 + 2*z1' + 2*z3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 14 * (1 + (1 + z1') + z3) + 2 * ((1 + (1 + z1') + z3) * (1 + (1 + z0') + z1)) + 4 * ((1 + (1 + z1') + z3) * (1 + (1 + z1') + z3)) + 4 * (1 + (1 + z0') + z1) + 10, s3 >= 0, s3 <= 2, s >= 0, s <= 1 + z1', z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 5 + z + 2*z2 + 2*z3 }-> 1 + s7 + s' :|: s7 >= 0, s7 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + 0 + (z - 1))) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + 0 + (z - 1)) + 10, s' >= 0, s' <= z2, z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 5 + 2*z' + z0'' + z1 }-> 1 + s8 + s'' :|: s8 >= 0, s8 <= 14 * (1 + 0 + (z' - 1)) + 2 * ((1 + 0 + (z' - 1)) * (1 + (1 + z0'') + z1)) + 4 * ((1 + 0 + (z' - 1)) * (1 + 0 + (z' - 1))) + 4 * (1 + (1 + z0'') + z1) + 10, s'' >= 0, s'' <= 0, z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 6 + z0 + z1 + 2*z2 + 2*z3 }-> 1 + s9 + s1 :|: s9 >= 0, s9 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + z0 + z1)) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + z0 + z1) + 10, s1 >= 0, s1 <= z2, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 3 + z' + 2*z2 }-> 1 + s27 :|: s27 >= 0, s27 <= 4 + 2 * z' + 7 * z2 + z2 * z' + 2 * (z2 * z2), z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 3 + 2*z'' + z1 }-> 1 + s28 :|: s28 >= 0, s28 <= 4 + 2 * z1 + 7 * z'' + z'' * z1 + 2 * (z'' * z''), z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s10 + MERGESORT(1 + (z' - 1) + 0) :|: s10 >= 0, s10 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s11 + MERGESORT(1 + (z' - 1) + 0) :|: s11 >= 0, s11 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s12 + MERGESORT(1 + (z' - 1) + 0) :|: s12 >= 0, s12 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s13 + MERGESORT(0) :|: s13 >= 0, s13 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s14 + MERGESORT(0) :|: s14 >= 0, s14 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s15 + MERGESORT(0) :|: s15 >= 0, s15 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s16 + MERGESORT(z') :|: s16 >= 0, s16 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s17 + MERGESORT(z') :|: s17 >= 0, s17 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s18 + MERGESORT(z') :|: s18 >= 0, s18 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s19 + MERGESORT(1 + (z'' - 1) + 0) :|: s19 >= 0, s19 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s20 + MERGESORT(0) :|: s20 >= 0, s20 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s21 + MERGESORT(z'') :|: s21 >= 0, s21 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s22 + MERGESORT(1 + (z'' - 1) + 0) :|: s22 >= 0, s22 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s23 + MERGESORT(z'') :|: s23 >= 0, s23 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s24 + MERGESORT(1 + (z'' - 1) + 0) :|: s24 >= 0, s24 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s25 + MERGESORT(0) :|: s25 >= 0, s25 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s26 + MERGESORT(z'') :|: s26 >= 0, s26 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), splitmerge(1 + z03 + (1 + z11 + z2''), 0, 0)) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), splitmerge(1 + z013 + (1 + z16 + z25), 0, 0)) + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(z'') :|: z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(0) :|: z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z017 + (1 + z18 + z27), 0, 0)) + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z019 + (1 + z19 + z28), 0, 0)) + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z07 + (1 + z13 + z22), 0, 0)) + MERGESORT(0) :|: z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z09 + (1 + z14 + z23), 0, 0)) + MERGESORT(z') :|: z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z015 + (1 + z17 + z26), 0, 0)) + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z05 + (1 + z12 + z21), 0, 0)) + MERGESORT(1 + (z' - 1) + 0) :|: z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 lteq(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> s38 :|: s38 >= 0, s38 <= 1 + (1 + z031) + z1 + (1 + (1 + z115) + z3), s5 >= 0, s5 <= 2, z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> s39 :|: s39 >= 0, s39 <= 1 + 0 + (z - 1) + (1 + z2 + z3), z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> s40 :|: s40 >= 0, s40 <= 1 + (1 + z032) + z1 + (1 + 0 + (z' - 1)), z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> s41 :|: s41 >= 0, s41 <= 1 + z0 + z1 + (1 + z2 + z3), z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + s43 :|: s43 >= 0, s43 <= z1 + z'', z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + s42 :|: s42 >= 0, s42 <= z' + z2, z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> s29 :|: s29 >= 0, s29 <= 1 + (z' - 1) + 0 + (1 + (z'' - 1) + 0), z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s30 :|: s30 >= 0, s30 <= 1 + (z' - 1) + 0 + 0, z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s31 :|: s31 >= 0, s31 <= 1 + (z' - 1) + 0 + 0, z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 0 + (1 + (z'' - 1) + 0), z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 0 + 0, z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= 0 + 0, z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 0 + (1 + (z'' - 1) + 0), z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> s36 :|: s36 >= 0, s36 <= 0 + 0, z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s37 :|: s37 >= 0, s37 <= 0 + 0, z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), splitmerge(1 + z023 + (1 + z111 + z210), 0, 0)) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 1 + (z'' - 1) + 0) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z027 + (1 + z113 + z212), 0, 0)) :|: z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z029 + (1 + z114 + z213), 0, 0)) :|: z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + z025 + (1 + z112 + z211), 0, 0)) :|: z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {splitmerge}, {mergesort}, {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] lteq: runtime: O(1) [0], size: O(1) [2] MERGE: runtime: O(n^1) [3 + z + 2*z'], size: O(n^2) [4 + 2*z + z*z' + 7*z' + 2*z'^2] MERGE[ITE]: runtime: O(n^1) [2 + z' + 2*z''], size: O(n^2) [10 + 4*z' + 2*z'*z'' + 14*z'' + 4*z''^2] merge[Ite]: runtime: O(1) [0], size: O(n^1) [z' + z''] merge: runtime: O(1) [0], size: O(n^1) [z + z'] splitmerge: runtime: ?, size: O(n^1) [14 + 29*z + 29*z' + 29*z''] ---------------------------------------- (71) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: splitmerge after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 9 + z0' + z1 + 2*z1' + 2*z3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 14 * (1 + (1 + z1') + z3) + 2 * ((1 + (1 + z1') + z3) * (1 + (1 + z0') + z1)) + 4 * ((1 + (1 + z1') + z3) * (1 + (1 + z1') + z3)) + 4 * (1 + (1 + z0') + z1) + 10, s3 >= 0, s3 <= 2, s >= 0, s <= 1 + z1', z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 5 + z + 2*z2 + 2*z3 }-> 1 + s7 + s' :|: s7 >= 0, s7 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + 0 + (z - 1))) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + 0 + (z - 1)) + 10, s' >= 0, s' <= z2, z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 5 + 2*z' + z0'' + z1 }-> 1 + s8 + s'' :|: s8 >= 0, s8 <= 14 * (1 + 0 + (z' - 1)) + 2 * ((1 + 0 + (z' - 1)) * (1 + (1 + z0'') + z1)) + 4 * ((1 + 0 + (z' - 1)) * (1 + 0 + (z' - 1))) + 4 * (1 + (1 + z0'') + z1) + 10, s'' >= 0, s'' <= 0, z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 6 + z0 + z1 + 2*z2 + 2*z3 }-> 1 + s9 + s1 :|: s9 >= 0, s9 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + z0 + z1)) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + z0 + z1) + 10, s1 >= 0, s1 <= z2, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 3 + z' + 2*z2 }-> 1 + s27 :|: s27 >= 0, s27 <= 4 + 2 * z' + 7 * z2 + z2 * z' + 2 * (z2 * z2), z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 3 + 2*z'' + z1 }-> 1 + s28 :|: s28 >= 0, s28 <= 4 + 2 * z1 + 7 * z'' + z'' * z1 + 2 * (z'' * z''), z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s10 + MERGESORT(1 + (z' - 1) + 0) :|: s10 >= 0, s10 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s11 + MERGESORT(1 + (z' - 1) + 0) :|: s11 >= 0, s11 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s12 + MERGESORT(1 + (z' - 1) + 0) :|: s12 >= 0, s12 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s13 + MERGESORT(0) :|: s13 >= 0, s13 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s14 + MERGESORT(0) :|: s14 >= 0, s14 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s15 + MERGESORT(0) :|: s15 >= 0, s15 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s16 + MERGESORT(z') :|: s16 >= 0, s16 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s17 + MERGESORT(z') :|: s17 >= 0, s17 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s18 + MERGESORT(z') :|: s18 >= 0, s18 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s19 + MERGESORT(1 + (z'' - 1) + 0) :|: s19 >= 0, s19 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s20 + MERGESORT(0) :|: s20 >= 0, s20 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s21 + MERGESORT(z'') :|: s21 >= 0, s21 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s22 + MERGESORT(1 + (z'' - 1) + 0) :|: s22 >= 0, s22 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s23 + MERGESORT(z'') :|: s23 >= 0, s23 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s24 + MERGESORT(1 + (z'' - 1) + 0) :|: s24 >= 0, s24 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s25 + MERGESORT(0) :|: s25 >= 0, s25 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s26 + MERGESORT(z'') :|: s26 >= 0, s26 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), splitmerge(1 + z03 + (1 + z11 + z2''), 0, 0)) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z01 + (1 + z1'' + z2'), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), splitmerge(1 + z013 + (1 + z16 + z25), 0, 0)) + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(z'') :|: z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 0) + MERGESORT(0) :|: z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(splitmerge(1 + z011 + (1 + z15 + z24), 0, 0), 1 + (z'' - 1) + 0) + MERGESORT(1 + (z'' - 1) + 0) :|: z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z017 + (1 + z18 + z27), 0, 0)) + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z019 + (1 + z19 + z28), 0, 0)) + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z07 + (1 + z13 + z22), 0, 0)) + MERGESORT(0) :|: z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(0, splitmerge(1 + z09 + (1 + z14 + z23), 0, 0)) + MERGESORT(z') :|: z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z015 + (1 + z17 + z26), 0, 0)) + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(1 + (z' - 1) + 0, splitmerge(1 + z05 + (1 + z12 + z21), 0, 0)) + MERGESORT(1 + (z' - 1) + 0) :|: z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 lteq(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> s38 :|: s38 >= 0, s38 <= 1 + (1 + z031) + z1 + (1 + (1 + z115) + z3), s5 >= 0, s5 <= 2, z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> s39 :|: s39 >= 0, s39 <= 1 + 0 + (z - 1) + (1 + z2 + z3), z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> s40 :|: s40 >= 0, s40 <= 1 + (1 + z032) + z1 + (1 + 0 + (z' - 1)), z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> s41 :|: s41 >= 0, s41 <= 1 + z0 + z1 + (1 + z2 + z3), z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + s43 :|: s43 >= 0, s43 <= z1 + z'', z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + s42 :|: s42 >= 0, s42 <= z' + z2, z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> s29 :|: s29 >= 0, s29 <= 1 + (z' - 1) + 0 + (1 + (z'' - 1) + 0), z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s30 :|: s30 >= 0, s30 <= 1 + (z' - 1) + 0 + 0, z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s31 :|: s31 >= 0, s31 <= 1 + (z' - 1) + 0 + 0, z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 0 + (1 + (z'' - 1) + 0), z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 0 + 0, z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= 0 + 0, z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 0 + (1 + (z'' - 1) + 0), z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> s36 :|: s36 >= 0, s36 <= 0 + 0, z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s37 :|: s37 >= 0, s37 <= 0 + 0, z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), splitmerge(1 + z023 + (1 + z111 + z210), 0, 0)) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 0) :|: z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(splitmerge(1 + z021 + (1 + z110 + z29), 0, 0), 1 + (z'' - 1) + 0) :|: z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z027 + (1 + z113 + z212), 0, 0)) :|: z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> merge(0, splitmerge(1 + z029 + (1 + z114 + z213), 0, 0)) :|: z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + z025 + (1 + z112 + z211), 0, 0)) :|: z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {mergesort}, {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] lteq: runtime: O(1) [0], size: O(1) [2] MERGE: runtime: O(n^1) [3 + z + 2*z'], size: O(n^2) [4 + 2*z + z*z' + 7*z' + 2*z'^2] MERGE[ITE]: runtime: O(n^1) [2 + z' + 2*z''], size: O(n^2) [10 + 4*z' + 2*z'*z'' + 14*z'' + 4*z''^2] merge[Ite]: runtime: O(1) [0], size: O(n^1) [z' + z''] merge: runtime: O(1) [0], size: O(n^1) [z + z'] splitmerge: runtime: O(1) [0], size: O(n^1) [14 + 29*z + 29*z' + 29*z''] ---------------------------------------- (73) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 9 + z0' + z1 + 2*z1' + 2*z3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 14 * (1 + (1 + z1') + z3) + 2 * ((1 + (1 + z1') + z3) * (1 + (1 + z0') + z1)) + 4 * ((1 + (1 + z1') + z3) * (1 + (1 + z1') + z3)) + 4 * (1 + (1 + z0') + z1) + 10, s3 >= 0, s3 <= 2, s >= 0, s <= 1 + z1', z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 5 + z + 2*z2 + 2*z3 }-> 1 + s7 + s' :|: s7 >= 0, s7 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + 0 + (z - 1))) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + 0 + (z - 1)) + 10, s' >= 0, s' <= z2, z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 5 + 2*z' + z0'' + z1 }-> 1 + s8 + s'' :|: s8 >= 0, s8 <= 14 * (1 + 0 + (z' - 1)) + 2 * ((1 + 0 + (z' - 1)) * (1 + (1 + z0'') + z1)) + 4 * ((1 + 0 + (z' - 1)) * (1 + 0 + (z' - 1))) + 4 * (1 + (1 + z0'') + z1) + 10, s'' >= 0, s'' <= 0, z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 6 + z0 + z1 + 2*z2 + 2*z3 }-> 1 + s9 + s1 :|: s9 >= 0, s9 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + z0 + z1)) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + z0 + z1) + 10, s1 >= 0, s1 <= z2, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 3 + z' + 2*z2 }-> 1 + s27 :|: s27 >= 0, s27 <= 4 + 2 * z' + 7 * z2 + z2 * z' + 2 * (z2 * z2), z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 3 + 2*z'' + z1 }-> 1 + s28 :|: s28 >= 0, s28 <= 4 + 2 * z1 + 7 * z'' + z'' * z1 + 2 * (z'' * z''), z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s10 + MERGESORT(1 + (z' - 1) + 0) :|: s10 >= 0, s10 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s11 + MERGESORT(1 + (z' - 1) + 0) :|: s11 >= 0, s11 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s12 + MERGESORT(1 + (z' - 1) + 0) :|: s12 >= 0, s12 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s13 + MERGESORT(0) :|: s13 >= 0, s13 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s14 + MERGESORT(0) :|: s14 >= 0, s14 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s15 + MERGESORT(0) :|: s15 >= 0, s15 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s16 + MERGESORT(z') :|: s16 >= 0, s16 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s17 + MERGESORT(z') :|: s17 >= 0, s17 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s18 + MERGESORT(z') :|: s18 >= 0, s18 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s19 + MERGESORT(1 + (z'' - 1) + 0) :|: s19 >= 0, s19 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s20 + MERGESORT(0) :|: s20 >= 0, s20 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s21 + MERGESORT(z'') :|: s21 >= 0, s21 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s22 + MERGESORT(1 + (z'' - 1) + 0) :|: s22 >= 0, s22 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s23 + MERGESORT(z'') :|: s23 >= 0, s23 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s24 + MERGESORT(1 + (z'' - 1) + 0) :|: s24 >= 0, s24 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s25 + MERGESORT(0) :|: s25 >= 0, s25 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s26 + MERGESORT(z'') :|: s26 >= 0, s26 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + s44 + 2*s45 }-> 1 + s46 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s44 >= 0, s44 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s45 >= 0, s45 <= 29 * (1 + z03 + (1 + z11 + z2'')) + 29 * 0 + 29 * 0 + 14, s46 >= 0, s46 <= 4 + 2 * s44 + 7 * s45 + s45 * s44 + 2 * (s45 * s45), z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + s47 + 2*z'' }-> 1 + s48 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s47 >= 0, s47 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s48 >= 0, s48 <= 4 + 2 * s47 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * s47 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + s49 }-> 1 + s50 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s49 >= 0, s49 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s50 >= 0, s50 <= 4 + 2 * s49 + 7 * 0 + 0 * s49 + 2 * (0 * 0), z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + s51 }-> 1 + s52 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s51 >= 0, s51 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s52 >= 0, s52 <= 4 + 2 * s51 + 7 * 0 + 0 * s51 + 2 * (0 * 0), z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s53 + z' }-> 1 + s54 + MERGESORT(1 + (z' - 1) + 0) :|: s53 >= 0, s53 <= 29 * (1 + z05 + (1 + z12 + z21)) + 29 * 0 + 29 * 0 + 14, s54 >= 0, s54 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * s53 + s53 * (1 + (z' - 1) + 0) + 2 * (s53 * s53), z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s55 }-> 1 + s56 + MERGESORT(0) :|: s55 >= 0, s55 <= 29 * (1 + z07 + (1 + z13 + z22)) + 29 * 0 + 29 * 0 + 14, s56 >= 0, s56 <= 4 + 2 * 0 + 7 * s55 + s55 * 0 + 2 * (s55 * s55), z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s57 }-> 1 + s58 + MERGESORT(z') :|: s57 >= 0, s57 <= 29 * (1 + z09 + (1 + z14 + z23)) + 29 * 0 + 29 * 0 + 14, s58 >= 0, s58 <= 4 + 2 * 0 + 7 * s57 + s57 * 0 + 2 * (s57 * s57), z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 4 + s59 + 2*s60 }-> 1 + s61 + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: s59 >= 0, s59 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s60 >= 0, s60 <= 29 * (1 + z013 + (1 + z16 + z25)) + 29 * 0 + 29 * 0 + 14, s61 >= 0, s61 <= 4 + 2 * s59 + 7 * s60 + s60 * s59 + 2 * (s60 * s60), z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 4 + s62 + 2*z'' }-> 1 + s63 + MERGESORT(1 + (z'' - 1) + 0) :|: s62 >= 0, s62 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s63 >= 0, s63 <= 4 + 2 * s62 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * s62 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + s64 }-> 1 + s65 + MERGESORT(0) :|: s64 >= 0, s64 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s65 >= 0, s65 <= 4 + 2 * s64 + 7 * 0 + 0 * s64 + 2 * (0 * 0), z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + s66 }-> 1 + s67 + MERGESORT(z'') :|: s66 >= 0, s66 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s67 >= 0, s67 <= 4 + 2 * s66 + 7 * 0 + 0 * s66 + 2 * (0 * 0), z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s68 + z' }-> 1 + s69 + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: s68 >= 0, s68 <= 29 * (1 + z015 + (1 + z17 + z26)) + 29 * 0 + 29 * 0 + 14, s69 >= 0, s69 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * s68 + s68 * (1 + (z' - 1) + 0) + 2 * (s68 * s68), z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s70 }-> 1 + s71 + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: s70 >= 0, s70 <= 29 * (1 + z017 + (1 + z18 + z27)) + 29 * 0 + 29 * 0 + 14, s71 >= 0, s71 <= 4 + 2 * 0 + 7 * s70 + s70 * 0 + 2 * (s70 * s70), z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s72 }-> 1 + s73 + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: s72 >= 0, s72 <= 29 * (1 + z019 + (1 + z19 + z28)) + 29 * 0 + 29 * 0 + 14, s73 >= 0, s73 <= 4 + 2 * 0 + 7 * s72 + s72 * 0 + 2 * (s72 * s72), z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 lteq(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> s38 :|: s38 >= 0, s38 <= 1 + (1 + z031) + z1 + (1 + (1 + z115) + z3), s5 >= 0, s5 <= 2, z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> s39 :|: s39 >= 0, s39 <= 1 + 0 + (z - 1) + (1 + z2 + z3), z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> s40 :|: s40 >= 0, s40 <= 1 + (1 + z032) + z1 + (1 + 0 + (z' - 1)), z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> s41 :|: s41 >= 0, s41 <= 1 + z0 + z1 + (1 + z2 + z3), z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + s43 :|: s43 >= 0, s43 <= z1 + z'', z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + s42 :|: s42 >= 0, s42 <= z' + z2, z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> s74 :|: s74 >= 0, s74 <= 29 * (1 + z0 + (1 + z1 + z2)) + 29 * 0 + 29 * 0 + 14, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> s29 :|: s29 >= 0, s29 <= 1 + (z' - 1) + 0 + (1 + (z'' - 1) + 0), z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s30 :|: s30 >= 0, s30 <= 1 + (z' - 1) + 0 + 0, z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s31 :|: s31 >= 0, s31 <= 1 + (z' - 1) + 0 + 0, z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 0 + (1 + (z'' - 1) + 0), z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 0 + 0, z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= 0 + 0, z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 0 + (1 + (z'' - 1) + 0), z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> s36 :|: s36 >= 0, s36 <= 0 + 0, z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s37 :|: s37 >= 0, s37 <= 0 + 0, z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s75 :|: s75 >= 0, s75 <= 29 * z1 + 29 * (1 + z0 + z'') + 29 * z' + 14, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> s78 :|: s76 >= 0, s76 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s77 >= 0, s77 <= 29 * (1 + z023 + (1 + z111 + z210)) + 29 * 0 + 29 * 0 + 14, s78 >= 0, s78 <= s76 + s77, z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s80 :|: s79 >= 0, s79 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s80 >= 0, s80 <= s79 + (1 + (z'' - 1) + 0), z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s82 :|: s81 >= 0, s81 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s82 >= 0, s82 <= s81 + 0, z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s84 :|: s83 >= 0, s83 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s84 >= 0, s84 <= s83 + 0, z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s86 :|: s85 >= 0, s85 <= 29 * (1 + z025 + (1 + z112 + z211)) + 29 * 0 + 29 * 0 + 14, s86 >= 0, s86 <= 1 + (z' - 1) + 0 + s85, z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> s88 :|: s87 >= 0, s87 <= 29 * (1 + z027 + (1 + z113 + z212)) + 29 * 0 + 29 * 0 + 14, s88 >= 0, s88 <= 0 + s87, z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> s90 :|: s89 >= 0, s89 <= 29 * (1 + z029 + (1 + z114 + z213)) + 29 * 0 + 29 * 0 + 14, s90 >= 0, s90 <= 0 + s89, z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {mergesort}, {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] lteq: runtime: O(1) [0], size: O(1) [2] MERGE: runtime: O(n^1) [3 + z + 2*z'], size: O(n^2) [4 + 2*z + z*z' + 7*z' + 2*z'^2] MERGE[ITE]: runtime: O(n^1) [2 + z' + 2*z''], size: O(n^2) [10 + 4*z' + 2*z'*z'' + 14*z'' + 4*z''^2] merge[Ite]: runtime: O(1) [0], size: O(n^1) [z' + z''] merge: runtime: O(1) [0], size: O(n^1) [z + z'] splitmerge: runtime: O(1) [0], size: O(n^1) [14 + 29*z + 29*z' + 29*z''] ---------------------------------------- (75) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: mergesort after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 37*z ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 9 + z0' + z1 + 2*z1' + 2*z3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 14 * (1 + (1 + z1') + z3) + 2 * ((1 + (1 + z1') + z3) * (1 + (1 + z0') + z1)) + 4 * ((1 + (1 + z1') + z3) * (1 + (1 + z1') + z3)) + 4 * (1 + (1 + z0') + z1) + 10, s3 >= 0, s3 <= 2, s >= 0, s <= 1 + z1', z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 5 + z + 2*z2 + 2*z3 }-> 1 + s7 + s' :|: s7 >= 0, s7 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + 0 + (z - 1))) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + 0 + (z - 1)) + 10, s' >= 0, s' <= z2, z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 5 + 2*z' + z0'' + z1 }-> 1 + s8 + s'' :|: s8 >= 0, s8 <= 14 * (1 + 0 + (z' - 1)) + 2 * ((1 + 0 + (z' - 1)) * (1 + (1 + z0'') + z1)) + 4 * ((1 + 0 + (z' - 1)) * (1 + 0 + (z' - 1))) + 4 * (1 + (1 + z0'') + z1) + 10, s'' >= 0, s'' <= 0, z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 6 + z0 + z1 + 2*z2 + 2*z3 }-> 1 + s9 + s1 :|: s9 >= 0, s9 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + z0 + z1)) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + z0 + z1) + 10, s1 >= 0, s1 <= z2, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 3 + z' + 2*z2 }-> 1 + s27 :|: s27 >= 0, s27 <= 4 + 2 * z' + 7 * z2 + z2 * z' + 2 * (z2 * z2), z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 3 + 2*z'' + z1 }-> 1 + s28 :|: s28 >= 0, s28 <= 4 + 2 * z1 + 7 * z'' + z'' * z1 + 2 * (z'' * z''), z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s10 + MERGESORT(1 + (z' - 1) + 0) :|: s10 >= 0, s10 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s11 + MERGESORT(1 + (z' - 1) + 0) :|: s11 >= 0, s11 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s12 + MERGESORT(1 + (z' - 1) + 0) :|: s12 >= 0, s12 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s13 + MERGESORT(0) :|: s13 >= 0, s13 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s14 + MERGESORT(0) :|: s14 >= 0, s14 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s15 + MERGESORT(0) :|: s15 >= 0, s15 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s16 + MERGESORT(z') :|: s16 >= 0, s16 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s17 + MERGESORT(z') :|: s17 >= 0, s17 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s18 + MERGESORT(z') :|: s18 >= 0, s18 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s19 + MERGESORT(1 + (z'' - 1) + 0) :|: s19 >= 0, s19 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s20 + MERGESORT(0) :|: s20 >= 0, s20 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s21 + MERGESORT(z'') :|: s21 >= 0, s21 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s22 + MERGESORT(1 + (z'' - 1) + 0) :|: s22 >= 0, s22 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s23 + MERGESORT(z'') :|: s23 >= 0, s23 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s24 + MERGESORT(1 + (z'' - 1) + 0) :|: s24 >= 0, s24 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s25 + MERGESORT(0) :|: s25 >= 0, s25 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s26 + MERGESORT(z'') :|: s26 >= 0, s26 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + s44 + 2*s45 }-> 1 + s46 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s44 >= 0, s44 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s45 >= 0, s45 <= 29 * (1 + z03 + (1 + z11 + z2'')) + 29 * 0 + 29 * 0 + 14, s46 >= 0, s46 <= 4 + 2 * s44 + 7 * s45 + s45 * s44 + 2 * (s45 * s45), z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + s47 + 2*z'' }-> 1 + s48 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s47 >= 0, s47 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s48 >= 0, s48 <= 4 + 2 * s47 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * s47 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + s49 }-> 1 + s50 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s49 >= 0, s49 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s50 >= 0, s50 <= 4 + 2 * s49 + 7 * 0 + 0 * s49 + 2 * (0 * 0), z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + s51 }-> 1 + s52 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s51 >= 0, s51 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s52 >= 0, s52 <= 4 + 2 * s51 + 7 * 0 + 0 * s51 + 2 * (0 * 0), z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s53 + z' }-> 1 + s54 + MERGESORT(1 + (z' - 1) + 0) :|: s53 >= 0, s53 <= 29 * (1 + z05 + (1 + z12 + z21)) + 29 * 0 + 29 * 0 + 14, s54 >= 0, s54 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * s53 + s53 * (1 + (z' - 1) + 0) + 2 * (s53 * s53), z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s55 }-> 1 + s56 + MERGESORT(0) :|: s55 >= 0, s55 <= 29 * (1 + z07 + (1 + z13 + z22)) + 29 * 0 + 29 * 0 + 14, s56 >= 0, s56 <= 4 + 2 * 0 + 7 * s55 + s55 * 0 + 2 * (s55 * s55), z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s57 }-> 1 + s58 + MERGESORT(z') :|: s57 >= 0, s57 <= 29 * (1 + z09 + (1 + z14 + z23)) + 29 * 0 + 29 * 0 + 14, s58 >= 0, s58 <= 4 + 2 * 0 + 7 * s57 + s57 * 0 + 2 * (s57 * s57), z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 4 + s59 + 2*s60 }-> 1 + s61 + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: s59 >= 0, s59 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s60 >= 0, s60 <= 29 * (1 + z013 + (1 + z16 + z25)) + 29 * 0 + 29 * 0 + 14, s61 >= 0, s61 <= 4 + 2 * s59 + 7 * s60 + s60 * s59 + 2 * (s60 * s60), z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 4 + s62 + 2*z'' }-> 1 + s63 + MERGESORT(1 + (z'' - 1) + 0) :|: s62 >= 0, s62 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s63 >= 0, s63 <= 4 + 2 * s62 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * s62 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + s64 }-> 1 + s65 + MERGESORT(0) :|: s64 >= 0, s64 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s65 >= 0, s65 <= 4 + 2 * s64 + 7 * 0 + 0 * s64 + 2 * (0 * 0), z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + s66 }-> 1 + s67 + MERGESORT(z'') :|: s66 >= 0, s66 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s67 >= 0, s67 <= 4 + 2 * s66 + 7 * 0 + 0 * s66 + 2 * (0 * 0), z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s68 + z' }-> 1 + s69 + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: s68 >= 0, s68 <= 29 * (1 + z015 + (1 + z17 + z26)) + 29 * 0 + 29 * 0 + 14, s69 >= 0, s69 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * s68 + s68 * (1 + (z' - 1) + 0) + 2 * (s68 * s68), z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s70 }-> 1 + s71 + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: s70 >= 0, s70 <= 29 * (1 + z017 + (1 + z18 + z27)) + 29 * 0 + 29 * 0 + 14, s71 >= 0, s71 <= 4 + 2 * 0 + 7 * s70 + s70 * 0 + 2 * (s70 * s70), z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s72 }-> 1 + s73 + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: s72 >= 0, s72 <= 29 * (1 + z019 + (1 + z19 + z28)) + 29 * 0 + 29 * 0 + 14, s73 >= 0, s73 <= 4 + 2 * 0 + 7 * s72 + s72 * 0 + 2 * (s72 * s72), z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 lteq(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> s38 :|: s38 >= 0, s38 <= 1 + (1 + z031) + z1 + (1 + (1 + z115) + z3), s5 >= 0, s5 <= 2, z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> s39 :|: s39 >= 0, s39 <= 1 + 0 + (z - 1) + (1 + z2 + z3), z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> s40 :|: s40 >= 0, s40 <= 1 + (1 + z032) + z1 + (1 + 0 + (z' - 1)), z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> s41 :|: s41 >= 0, s41 <= 1 + z0 + z1 + (1 + z2 + z3), z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + s43 :|: s43 >= 0, s43 <= z1 + z'', z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + s42 :|: s42 >= 0, s42 <= z' + z2, z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> s74 :|: s74 >= 0, s74 <= 29 * (1 + z0 + (1 + z1 + z2)) + 29 * 0 + 29 * 0 + 14, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> s29 :|: s29 >= 0, s29 <= 1 + (z' - 1) + 0 + (1 + (z'' - 1) + 0), z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s30 :|: s30 >= 0, s30 <= 1 + (z' - 1) + 0 + 0, z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s31 :|: s31 >= 0, s31 <= 1 + (z' - 1) + 0 + 0, z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 0 + (1 + (z'' - 1) + 0), z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 0 + 0, z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= 0 + 0, z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 0 + (1 + (z'' - 1) + 0), z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> s36 :|: s36 >= 0, s36 <= 0 + 0, z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s37 :|: s37 >= 0, s37 <= 0 + 0, z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s75 :|: s75 >= 0, s75 <= 29 * z1 + 29 * (1 + z0 + z'') + 29 * z' + 14, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> s78 :|: s76 >= 0, s76 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s77 >= 0, s77 <= 29 * (1 + z023 + (1 + z111 + z210)) + 29 * 0 + 29 * 0 + 14, s78 >= 0, s78 <= s76 + s77, z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s80 :|: s79 >= 0, s79 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s80 >= 0, s80 <= s79 + (1 + (z'' - 1) + 0), z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s82 :|: s81 >= 0, s81 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s82 >= 0, s82 <= s81 + 0, z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s84 :|: s83 >= 0, s83 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s84 >= 0, s84 <= s83 + 0, z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s86 :|: s85 >= 0, s85 <= 29 * (1 + z025 + (1 + z112 + z211)) + 29 * 0 + 29 * 0 + 14, s86 >= 0, s86 <= 1 + (z' - 1) + 0 + s85, z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> s88 :|: s87 >= 0, s87 <= 29 * (1 + z027 + (1 + z113 + z212)) + 29 * 0 + 29 * 0 + 14, s88 >= 0, s88 <= 0 + s87, z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> s90 :|: s89 >= 0, s89 <= 29 * (1 + z029 + (1 + z114 + z213)) + 29 * 0 + 29 * 0 + 14, s90 >= 0, s90 <= 0 + s89, z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {mergesort}, {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] lteq: runtime: O(1) [0], size: O(1) [2] MERGE: runtime: O(n^1) [3 + z + 2*z'], size: O(n^2) [4 + 2*z + z*z' + 7*z' + 2*z'^2] MERGE[ITE]: runtime: O(n^1) [2 + z' + 2*z''], size: O(n^2) [10 + 4*z' + 2*z'*z'' + 14*z'' + 4*z''^2] merge[Ite]: runtime: O(1) [0], size: O(n^1) [z' + z''] merge: runtime: O(1) [0], size: O(n^1) [z + z'] splitmerge: runtime: O(1) [0], size: O(n^1) [14 + 29*z + 29*z' + 29*z''] mergesort: runtime: ?, size: O(n^1) [37*z] ---------------------------------------- (77) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: mergesort after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 9 + z0' + z1 + 2*z1' + 2*z3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 14 * (1 + (1 + z1') + z3) + 2 * ((1 + (1 + z1') + z3) * (1 + (1 + z0') + z1)) + 4 * ((1 + (1 + z1') + z3) * (1 + (1 + z1') + z3)) + 4 * (1 + (1 + z0') + z1) + 10, s3 >= 0, s3 <= 2, s >= 0, s <= 1 + z1', z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 5 + z + 2*z2 + 2*z3 }-> 1 + s7 + s' :|: s7 >= 0, s7 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + 0 + (z - 1))) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + 0 + (z - 1)) + 10, s' >= 0, s' <= z2, z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 5 + 2*z' + z0'' + z1 }-> 1 + s8 + s'' :|: s8 >= 0, s8 <= 14 * (1 + 0 + (z' - 1)) + 2 * ((1 + 0 + (z' - 1)) * (1 + (1 + z0'') + z1)) + 4 * ((1 + 0 + (z' - 1)) * (1 + 0 + (z' - 1))) + 4 * (1 + (1 + z0'') + z1) + 10, s'' >= 0, s'' <= 0, z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 6 + z0 + z1 + 2*z2 + 2*z3 }-> 1 + s9 + s1 :|: s9 >= 0, s9 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + z0 + z1)) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + z0 + z1) + 10, s1 >= 0, s1 <= z2, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 3 + z' + 2*z2 }-> 1 + s27 :|: s27 >= 0, s27 <= 4 + 2 * z' + 7 * z2 + z2 * z' + 2 * (z2 * z2), z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 3 + 2*z'' + z1 }-> 1 + s28 :|: s28 >= 0, s28 <= 4 + 2 * z1 + 7 * z'' + z'' * z1 + 2 * (z'' * z''), z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s10 + MERGESORT(1 + (z' - 1) + 0) :|: s10 >= 0, s10 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s11 + MERGESORT(1 + (z' - 1) + 0) :|: s11 >= 0, s11 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s12 + MERGESORT(1 + (z' - 1) + 0) :|: s12 >= 0, s12 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s13 + MERGESORT(0) :|: s13 >= 0, s13 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s14 + MERGESORT(0) :|: s14 >= 0, s14 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s15 + MERGESORT(0) :|: s15 >= 0, s15 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s16 + MERGESORT(z') :|: s16 >= 0, s16 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s17 + MERGESORT(z') :|: s17 >= 0, s17 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s18 + MERGESORT(z') :|: s18 >= 0, s18 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s19 + MERGESORT(1 + (z'' - 1) + 0) :|: s19 >= 0, s19 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s20 + MERGESORT(0) :|: s20 >= 0, s20 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s21 + MERGESORT(z'') :|: s21 >= 0, s21 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s22 + MERGESORT(1 + (z'' - 1) + 0) :|: s22 >= 0, s22 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s23 + MERGESORT(z'') :|: s23 >= 0, s23 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s24 + MERGESORT(1 + (z'' - 1) + 0) :|: s24 >= 0, s24 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s25 + MERGESORT(0) :|: s25 >= 0, s25 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s26 + MERGESORT(z'') :|: s26 >= 0, s26 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + s44 + 2*s45 }-> 1 + s46 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s44 >= 0, s44 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s45 >= 0, s45 <= 29 * (1 + z03 + (1 + z11 + z2'')) + 29 * 0 + 29 * 0 + 14, s46 >= 0, s46 <= 4 + 2 * s44 + 7 * s45 + s45 * s44 + 2 * (s45 * s45), z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + s47 + 2*z'' }-> 1 + s48 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s47 >= 0, s47 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s48 >= 0, s48 <= 4 + 2 * s47 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * s47 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + s49 }-> 1 + s50 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s49 >= 0, s49 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s50 >= 0, s50 <= 4 + 2 * s49 + 7 * 0 + 0 * s49 + 2 * (0 * 0), z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + s51 }-> 1 + s52 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s51 >= 0, s51 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s52 >= 0, s52 <= 4 + 2 * s51 + 7 * 0 + 0 * s51 + 2 * (0 * 0), z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s53 + z' }-> 1 + s54 + MERGESORT(1 + (z' - 1) + 0) :|: s53 >= 0, s53 <= 29 * (1 + z05 + (1 + z12 + z21)) + 29 * 0 + 29 * 0 + 14, s54 >= 0, s54 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * s53 + s53 * (1 + (z' - 1) + 0) + 2 * (s53 * s53), z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s55 }-> 1 + s56 + MERGESORT(0) :|: s55 >= 0, s55 <= 29 * (1 + z07 + (1 + z13 + z22)) + 29 * 0 + 29 * 0 + 14, s56 >= 0, s56 <= 4 + 2 * 0 + 7 * s55 + s55 * 0 + 2 * (s55 * s55), z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s57 }-> 1 + s58 + MERGESORT(z') :|: s57 >= 0, s57 <= 29 * (1 + z09 + (1 + z14 + z23)) + 29 * 0 + 29 * 0 + 14, s58 >= 0, s58 <= 4 + 2 * 0 + 7 * s57 + s57 * 0 + 2 * (s57 * s57), z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 4 + s59 + 2*s60 }-> 1 + s61 + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: s59 >= 0, s59 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s60 >= 0, s60 <= 29 * (1 + z013 + (1 + z16 + z25)) + 29 * 0 + 29 * 0 + 14, s61 >= 0, s61 <= 4 + 2 * s59 + 7 * s60 + s60 * s59 + 2 * (s60 * s60), z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 4 + s62 + 2*z'' }-> 1 + s63 + MERGESORT(1 + (z'' - 1) + 0) :|: s62 >= 0, s62 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s63 >= 0, s63 <= 4 + 2 * s62 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * s62 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + s64 }-> 1 + s65 + MERGESORT(0) :|: s64 >= 0, s64 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s65 >= 0, s65 <= 4 + 2 * s64 + 7 * 0 + 0 * s64 + 2 * (0 * 0), z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + s66 }-> 1 + s67 + MERGESORT(z'') :|: s66 >= 0, s66 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s67 >= 0, s67 <= 4 + 2 * s66 + 7 * 0 + 0 * s66 + 2 * (0 * 0), z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s68 + z' }-> 1 + s69 + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: s68 >= 0, s68 <= 29 * (1 + z015 + (1 + z17 + z26)) + 29 * 0 + 29 * 0 + 14, s69 >= 0, s69 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * s68 + s68 * (1 + (z' - 1) + 0) + 2 * (s68 * s68), z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s70 }-> 1 + s71 + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: s70 >= 0, s70 <= 29 * (1 + z017 + (1 + z18 + z27)) + 29 * 0 + 29 * 0 + 14, s71 >= 0, s71 <= 4 + 2 * 0 + 7 * s70 + s70 * 0 + 2 * (s70 * s70), z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s72 }-> 1 + s73 + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: s72 >= 0, s72 <= 29 * (1 + z019 + (1 + z19 + z28)) + 29 * 0 + 29 * 0 + 14, s73 >= 0, s73 <= 4 + 2 * 0 + 7 * s72 + s72 * 0 + 2 * (s72 * s72), z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 lteq(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> s38 :|: s38 >= 0, s38 <= 1 + (1 + z031) + z1 + (1 + (1 + z115) + z3), s5 >= 0, s5 <= 2, z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> s39 :|: s39 >= 0, s39 <= 1 + 0 + (z - 1) + (1 + z2 + z3), z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> s40 :|: s40 >= 0, s40 <= 1 + (1 + z032) + z1 + (1 + 0 + (z' - 1)), z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> s41 :|: s41 >= 0, s41 <= 1 + z0 + z1 + (1 + z2 + z3), z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + s43 :|: s43 >= 0, s43 <= z1 + z'', z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + s42 :|: s42 >= 0, s42 <= z' + z2, z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> s74 :|: s74 >= 0, s74 <= 29 * (1 + z0 + (1 + z1 + z2)) + 29 * 0 + 29 * 0 + 14, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> s29 :|: s29 >= 0, s29 <= 1 + (z' - 1) + 0 + (1 + (z'' - 1) + 0), z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s30 :|: s30 >= 0, s30 <= 1 + (z' - 1) + 0 + 0, z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s31 :|: s31 >= 0, s31 <= 1 + (z' - 1) + 0 + 0, z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 0 + (1 + (z'' - 1) + 0), z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 0 + 0, z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= 0 + 0, z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 0 + (1 + (z'' - 1) + 0), z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> s36 :|: s36 >= 0, s36 <= 0 + 0, z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s37 :|: s37 >= 0, s37 <= 0 + 0, z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s75 :|: s75 >= 0, s75 <= 29 * z1 + 29 * (1 + z0 + z'') + 29 * z' + 14, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> s78 :|: s76 >= 0, s76 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s77 >= 0, s77 <= 29 * (1 + z023 + (1 + z111 + z210)) + 29 * 0 + 29 * 0 + 14, s78 >= 0, s78 <= s76 + s77, z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s80 :|: s79 >= 0, s79 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s80 >= 0, s80 <= s79 + (1 + (z'' - 1) + 0), z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s82 :|: s81 >= 0, s81 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s82 >= 0, s82 <= s81 + 0, z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s84 :|: s83 >= 0, s83 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s84 >= 0, s84 <= s83 + 0, z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s86 :|: s85 >= 0, s85 <= 29 * (1 + z025 + (1 + z112 + z211)) + 29 * 0 + 29 * 0 + 14, s86 >= 0, s86 <= 1 + (z' - 1) + 0 + s85, z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> s88 :|: s87 >= 0, s87 <= 29 * (1 + z027 + (1 + z113 + z212)) + 29 * 0 + 29 * 0 + 14, s88 >= 0, s88 <= 0 + s87, z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> s90 :|: s89 >= 0, s89 <= 29 * (1 + z029 + (1 + z114 + z213)) + 29 * 0 + 29 * 0 + 14, s90 >= 0, s90 <= 0 + s89, z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] lteq: runtime: O(1) [0], size: O(1) [2] MERGE: runtime: O(n^1) [3 + z + 2*z'], size: O(n^2) [4 + 2*z + z*z' + 7*z' + 2*z'^2] MERGE[ITE]: runtime: O(n^1) [2 + z' + 2*z''], size: O(n^2) [10 + 4*z' + 2*z'*z'' + 14*z'' + 4*z''^2] merge[Ite]: runtime: O(1) [0], size: O(n^1) [z' + z''] merge: runtime: O(1) [0], size: O(n^1) [z + z'] splitmerge: runtime: O(1) [0], size: O(n^1) [14 + 29*z + 29*z' + 29*z''] mergesort: runtime: O(1) [0], size: O(n^1) [37*z] ---------------------------------------- (79) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (80) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 9 + z0' + z1 + 2*z1' + 2*z3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 14 * (1 + (1 + z1') + z3) + 2 * ((1 + (1 + z1') + z3) * (1 + (1 + z0') + z1)) + 4 * ((1 + (1 + z1') + z3) * (1 + (1 + z1') + z3)) + 4 * (1 + (1 + z0') + z1) + 10, s3 >= 0, s3 <= 2, s >= 0, s <= 1 + z1', z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 5 + z + 2*z2 + 2*z3 }-> 1 + s7 + s' :|: s7 >= 0, s7 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + 0 + (z - 1))) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + 0 + (z - 1)) + 10, s' >= 0, s' <= z2, z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 5 + 2*z' + z0'' + z1 }-> 1 + s8 + s'' :|: s8 >= 0, s8 <= 14 * (1 + 0 + (z' - 1)) + 2 * ((1 + 0 + (z' - 1)) * (1 + (1 + z0'') + z1)) + 4 * ((1 + 0 + (z' - 1)) * (1 + 0 + (z' - 1))) + 4 * (1 + (1 + z0'') + z1) + 10, s'' >= 0, s'' <= 0, z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 6 + z0 + z1 + 2*z2 + 2*z3 }-> 1 + s9 + s1 :|: s9 >= 0, s9 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + z0 + z1)) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + z0 + z1) + 10, s1 >= 0, s1 <= z2, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 3 + z' + 2*z2 }-> 1 + s27 :|: s27 >= 0, s27 <= 4 + 2 * z' + 7 * z2 + z2 * z' + 2 * (z2 * z2), z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 3 + 2*z'' + z1 }-> 1 + s28 :|: s28 >= 0, s28 <= 4 + 2 * z1 + 7 * z'' + z'' * z1 + 2 * (z'' * z''), z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s10 + MERGESORT(1 + (z' - 1) + 0) :|: s10 >= 0, s10 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s11 + MERGESORT(1 + (z' - 1) + 0) :|: s11 >= 0, s11 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s12 + MERGESORT(1 + (z' - 1) + 0) :|: s12 >= 0, s12 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s13 + MERGESORT(0) :|: s13 >= 0, s13 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s14 + MERGESORT(0) :|: s14 >= 0, s14 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s15 + MERGESORT(0) :|: s15 >= 0, s15 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s16 + MERGESORT(z') :|: s16 >= 0, s16 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s17 + MERGESORT(z') :|: s17 >= 0, s17 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s18 + MERGESORT(z') :|: s18 >= 0, s18 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s19 + MERGESORT(1 + (z'' - 1) + 0) :|: s19 >= 0, s19 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s20 + MERGESORT(0) :|: s20 >= 0, s20 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s21 + MERGESORT(z'') :|: s21 >= 0, s21 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s22 + MERGESORT(1 + (z'' - 1) + 0) :|: s22 >= 0, s22 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s23 + MERGESORT(z'') :|: s23 >= 0, s23 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s24 + MERGESORT(1 + (z'' - 1) + 0) :|: s24 >= 0, s24 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s25 + MERGESORT(0) :|: s25 >= 0, s25 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s26 + MERGESORT(z'') :|: s26 >= 0, s26 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + s44 + 2*s45 }-> 1 + s46 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s44 >= 0, s44 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s45 >= 0, s45 <= 29 * (1 + z03 + (1 + z11 + z2'')) + 29 * 0 + 29 * 0 + 14, s46 >= 0, s46 <= 4 + 2 * s44 + 7 * s45 + s45 * s44 + 2 * (s45 * s45), z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + s47 + 2*z'' }-> 1 + s48 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s47 >= 0, s47 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s48 >= 0, s48 <= 4 + 2 * s47 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * s47 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + s49 }-> 1 + s50 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s49 >= 0, s49 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s50 >= 0, s50 <= 4 + 2 * s49 + 7 * 0 + 0 * s49 + 2 * (0 * 0), z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + s51 }-> 1 + s52 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s51 >= 0, s51 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s52 >= 0, s52 <= 4 + 2 * s51 + 7 * 0 + 0 * s51 + 2 * (0 * 0), z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s53 + z' }-> 1 + s54 + MERGESORT(1 + (z' - 1) + 0) :|: s53 >= 0, s53 <= 29 * (1 + z05 + (1 + z12 + z21)) + 29 * 0 + 29 * 0 + 14, s54 >= 0, s54 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * s53 + s53 * (1 + (z' - 1) + 0) + 2 * (s53 * s53), z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s55 }-> 1 + s56 + MERGESORT(0) :|: s55 >= 0, s55 <= 29 * (1 + z07 + (1 + z13 + z22)) + 29 * 0 + 29 * 0 + 14, s56 >= 0, s56 <= 4 + 2 * 0 + 7 * s55 + s55 * 0 + 2 * (s55 * s55), z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s57 }-> 1 + s58 + MERGESORT(z') :|: s57 >= 0, s57 <= 29 * (1 + z09 + (1 + z14 + z23)) + 29 * 0 + 29 * 0 + 14, s58 >= 0, s58 <= 4 + 2 * 0 + 7 * s57 + s57 * 0 + 2 * (s57 * s57), z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 4 + s59 + 2*s60 }-> 1 + s61 + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: s59 >= 0, s59 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s60 >= 0, s60 <= 29 * (1 + z013 + (1 + z16 + z25)) + 29 * 0 + 29 * 0 + 14, s61 >= 0, s61 <= 4 + 2 * s59 + 7 * s60 + s60 * s59 + 2 * (s60 * s60), z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 4 + s62 + 2*z'' }-> 1 + s63 + MERGESORT(1 + (z'' - 1) + 0) :|: s62 >= 0, s62 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s63 >= 0, s63 <= 4 + 2 * s62 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * s62 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + s64 }-> 1 + s65 + MERGESORT(0) :|: s64 >= 0, s64 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s65 >= 0, s65 <= 4 + 2 * s64 + 7 * 0 + 0 * s64 + 2 * (0 * 0), z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + s66 }-> 1 + s67 + MERGESORT(z'') :|: s66 >= 0, s66 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s67 >= 0, s67 <= 4 + 2 * s66 + 7 * 0 + 0 * s66 + 2 * (0 * 0), z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s68 + z' }-> 1 + s69 + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: s68 >= 0, s68 <= 29 * (1 + z015 + (1 + z17 + z26)) + 29 * 0 + 29 * 0 + 14, s69 >= 0, s69 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * s68 + s68 * (1 + (z' - 1) + 0) + 2 * (s68 * s68), z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s70 }-> 1 + s71 + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: s70 >= 0, s70 <= 29 * (1 + z017 + (1 + z18 + z27)) + 29 * 0 + 29 * 0 + 14, s71 >= 0, s71 <= 4 + 2 * 0 + 7 * s70 + s70 * 0 + 2 * (s70 * s70), z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s72 }-> 1 + s73 + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: s72 >= 0, s72 <= 29 * (1 + z019 + (1 + z19 + z28)) + 29 * 0 + 29 * 0 + 14, s73 >= 0, s73 <= 4 + 2 * 0 + 7 * s72 + s72 * 0 + 2 * (s72 * s72), z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 lteq(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> s38 :|: s38 >= 0, s38 <= 1 + (1 + z031) + z1 + (1 + (1 + z115) + z3), s5 >= 0, s5 <= 2, z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> s39 :|: s39 >= 0, s39 <= 1 + 0 + (z - 1) + (1 + z2 + z3), z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> s40 :|: s40 >= 0, s40 <= 1 + (1 + z032) + z1 + (1 + 0 + (z' - 1)), z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> s41 :|: s41 >= 0, s41 <= 1 + z0 + z1 + (1 + z2 + z3), z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + s43 :|: s43 >= 0, s43 <= z1 + z'', z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + s42 :|: s42 >= 0, s42 <= z' + z2, z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> s74 :|: s74 >= 0, s74 <= 29 * (1 + z0 + (1 + z1 + z2)) + 29 * 0 + 29 * 0 + 14, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> s29 :|: s29 >= 0, s29 <= 1 + (z' - 1) + 0 + (1 + (z'' - 1) + 0), z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s30 :|: s30 >= 0, s30 <= 1 + (z' - 1) + 0 + 0, z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s31 :|: s31 >= 0, s31 <= 1 + (z' - 1) + 0 + 0, z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 0 + (1 + (z'' - 1) + 0), z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 0 + 0, z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= 0 + 0, z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 0 + (1 + (z'' - 1) + 0), z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> s36 :|: s36 >= 0, s36 <= 0 + 0, z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s37 :|: s37 >= 0, s37 <= 0 + 0, z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s75 :|: s75 >= 0, s75 <= 29 * z1 + 29 * (1 + z0 + z'') + 29 * z' + 14, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> s78 :|: s76 >= 0, s76 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s77 >= 0, s77 <= 29 * (1 + z023 + (1 + z111 + z210)) + 29 * 0 + 29 * 0 + 14, s78 >= 0, s78 <= s76 + s77, z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s80 :|: s79 >= 0, s79 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s80 >= 0, s80 <= s79 + (1 + (z'' - 1) + 0), z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s82 :|: s81 >= 0, s81 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s82 >= 0, s82 <= s81 + 0, z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s84 :|: s83 >= 0, s83 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s84 >= 0, s84 <= s83 + 0, z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s86 :|: s85 >= 0, s85 <= 29 * (1 + z025 + (1 + z112 + z211)) + 29 * 0 + 29 * 0 + 14, s86 >= 0, s86 <= 1 + (z' - 1) + 0 + s85, z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> s88 :|: s87 >= 0, s87 <= 29 * (1 + z027 + (1 + z113 + z212)) + 29 * 0 + 29 * 0 + 14, s88 >= 0, s88 <= 0 + s87, z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> s90 :|: s89 >= 0, s89 <= 29 * (1 + z029 + (1 + z114 + z213)) + 29 * 0 + 29 * 0 + 14, s90 >= 0, s90 <= 0 + s89, z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] lteq: runtime: O(1) [0], size: O(1) [2] MERGE: runtime: O(n^1) [3 + z + 2*z'], size: O(n^2) [4 + 2*z + z*z' + 7*z' + 2*z'^2] MERGE[ITE]: runtime: O(n^1) [2 + z' + 2*z''], size: O(n^2) [10 + 4*z' + 2*z'*z'' + 14*z'' + 4*z''^2] merge[Ite]: runtime: O(1) [0], size: O(n^1) [z' + z''] merge: runtime: O(1) [0], size: O(n^1) [z + z'] splitmerge: runtime: O(1) [0], size: O(n^1) [14 + 29*z + 29*z' + 29*z''] mergesort: runtime: O(1) [0], size: O(n^1) [37*z] ---------------------------------------- (81) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: MERGESORT after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed SIZE bound using CoFloCo for: SPLITMERGE after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (82) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 9 + z0' + z1 + 2*z1' + 2*z3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 14 * (1 + (1 + z1') + z3) + 2 * ((1 + (1 + z1') + z3) * (1 + (1 + z0') + z1)) + 4 * ((1 + (1 + z1') + z3) * (1 + (1 + z1') + z3)) + 4 * (1 + (1 + z0') + z1) + 10, s3 >= 0, s3 <= 2, s >= 0, s <= 1 + z1', z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 5 + z + 2*z2 + 2*z3 }-> 1 + s7 + s' :|: s7 >= 0, s7 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + 0 + (z - 1))) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + 0 + (z - 1)) + 10, s' >= 0, s' <= z2, z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 5 + 2*z' + z0'' + z1 }-> 1 + s8 + s'' :|: s8 >= 0, s8 <= 14 * (1 + 0 + (z' - 1)) + 2 * ((1 + 0 + (z' - 1)) * (1 + (1 + z0'') + z1)) + 4 * ((1 + 0 + (z' - 1)) * (1 + 0 + (z' - 1))) + 4 * (1 + (1 + z0'') + z1) + 10, s'' >= 0, s'' <= 0, z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 6 + z0 + z1 + 2*z2 + 2*z3 }-> 1 + s9 + s1 :|: s9 >= 0, s9 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + z0 + z1)) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + z0 + z1) + 10, s1 >= 0, s1 <= z2, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 3 + z' + 2*z2 }-> 1 + s27 :|: s27 >= 0, s27 <= 4 + 2 * z' + 7 * z2 + z2 * z' + 2 * (z2 * z2), z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 3 + 2*z'' + z1 }-> 1 + s28 :|: s28 >= 0, s28 <= 4 + 2 * z1 + 7 * z'' + z'' * z1 + 2 * (z'' * z''), z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s10 + MERGESORT(1 + (z' - 1) + 0) :|: s10 >= 0, s10 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s11 + MERGESORT(1 + (z' - 1) + 0) :|: s11 >= 0, s11 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s12 + MERGESORT(1 + (z' - 1) + 0) :|: s12 >= 0, s12 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s13 + MERGESORT(0) :|: s13 >= 0, s13 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s14 + MERGESORT(0) :|: s14 >= 0, s14 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s15 + MERGESORT(0) :|: s15 >= 0, s15 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s16 + MERGESORT(z') :|: s16 >= 0, s16 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s17 + MERGESORT(z') :|: s17 >= 0, s17 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s18 + MERGESORT(z') :|: s18 >= 0, s18 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s19 + MERGESORT(1 + (z'' - 1) + 0) :|: s19 >= 0, s19 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s20 + MERGESORT(0) :|: s20 >= 0, s20 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s21 + MERGESORT(z'') :|: s21 >= 0, s21 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s22 + MERGESORT(1 + (z'' - 1) + 0) :|: s22 >= 0, s22 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s23 + MERGESORT(z'') :|: s23 >= 0, s23 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s24 + MERGESORT(1 + (z'' - 1) + 0) :|: s24 >= 0, s24 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s25 + MERGESORT(0) :|: s25 >= 0, s25 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s26 + MERGESORT(z'') :|: s26 >= 0, s26 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + s44 + 2*s45 }-> 1 + s46 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s44 >= 0, s44 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s45 >= 0, s45 <= 29 * (1 + z03 + (1 + z11 + z2'')) + 29 * 0 + 29 * 0 + 14, s46 >= 0, s46 <= 4 + 2 * s44 + 7 * s45 + s45 * s44 + 2 * (s45 * s45), z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + s47 + 2*z'' }-> 1 + s48 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s47 >= 0, s47 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s48 >= 0, s48 <= 4 + 2 * s47 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * s47 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + s49 }-> 1 + s50 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s49 >= 0, s49 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s50 >= 0, s50 <= 4 + 2 * s49 + 7 * 0 + 0 * s49 + 2 * (0 * 0), z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + s51 }-> 1 + s52 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s51 >= 0, s51 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s52 >= 0, s52 <= 4 + 2 * s51 + 7 * 0 + 0 * s51 + 2 * (0 * 0), z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s53 + z' }-> 1 + s54 + MERGESORT(1 + (z' - 1) + 0) :|: s53 >= 0, s53 <= 29 * (1 + z05 + (1 + z12 + z21)) + 29 * 0 + 29 * 0 + 14, s54 >= 0, s54 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * s53 + s53 * (1 + (z' - 1) + 0) + 2 * (s53 * s53), z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s55 }-> 1 + s56 + MERGESORT(0) :|: s55 >= 0, s55 <= 29 * (1 + z07 + (1 + z13 + z22)) + 29 * 0 + 29 * 0 + 14, s56 >= 0, s56 <= 4 + 2 * 0 + 7 * s55 + s55 * 0 + 2 * (s55 * s55), z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s57 }-> 1 + s58 + MERGESORT(z') :|: s57 >= 0, s57 <= 29 * (1 + z09 + (1 + z14 + z23)) + 29 * 0 + 29 * 0 + 14, s58 >= 0, s58 <= 4 + 2 * 0 + 7 * s57 + s57 * 0 + 2 * (s57 * s57), z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 4 + s59 + 2*s60 }-> 1 + s61 + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: s59 >= 0, s59 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s60 >= 0, s60 <= 29 * (1 + z013 + (1 + z16 + z25)) + 29 * 0 + 29 * 0 + 14, s61 >= 0, s61 <= 4 + 2 * s59 + 7 * s60 + s60 * s59 + 2 * (s60 * s60), z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 4 + s62 + 2*z'' }-> 1 + s63 + MERGESORT(1 + (z'' - 1) + 0) :|: s62 >= 0, s62 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s63 >= 0, s63 <= 4 + 2 * s62 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * s62 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + s64 }-> 1 + s65 + MERGESORT(0) :|: s64 >= 0, s64 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s65 >= 0, s65 <= 4 + 2 * s64 + 7 * 0 + 0 * s64 + 2 * (0 * 0), z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + s66 }-> 1 + s67 + MERGESORT(z'') :|: s66 >= 0, s66 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s67 >= 0, s67 <= 4 + 2 * s66 + 7 * 0 + 0 * s66 + 2 * (0 * 0), z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s68 + z' }-> 1 + s69 + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: s68 >= 0, s68 <= 29 * (1 + z015 + (1 + z17 + z26)) + 29 * 0 + 29 * 0 + 14, s69 >= 0, s69 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * s68 + s68 * (1 + (z' - 1) + 0) + 2 * (s68 * s68), z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s70 }-> 1 + s71 + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: s70 >= 0, s70 <= 29 * (1 + z017 + (1 + z18 + z27)) + 29 * 0 + 29 * 0 + 14, s71 >= 0, s71 <= 4 + 2 * 0 + 7 * s70 + s70 * 0 + 2 * (s70 * s70), z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s72 }-> 1 + s73 + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: s72 >= 0, s72 <= 29 * (1 + z019 + (1 + z19 + z28)) + 29 * 0 + 29 * 0 + 14, s73 >= 0, s73 <= 4 + 2 * 0 + 7 * s72 + s72 * 0 + 2 * (s72 * s72), z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 lteq(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> s38 :|: s38 >= 0, s38 <= 1 + (1 + z031) + z1 + (1 + (1 + z115) + z3), s5 >= 0, s5 <= 2, z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> s39 :|: s39 >= 0, s39 <= 1 + 0 + (z - 1) + (1 + z2 + z3), z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> s40 :|: s40 >= 0, s40 <= 1 + (1 + z032) + z1 + (1 + 0 + (z' - 1)), z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> s41 :|: s41 >= 0, s41 <= 1 + z0 + z1 + (1 + z2 + z3), z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + s43 :|: s43 >= 0, s43 <= z1 + z'', z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + s42 :|: s42 >= 0, s42 <= z' + z2, z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> s74 :|: s74 >= 0, s74 <= 29 * (1 + z0 + (1 + z1 + z2)) + 29 * 0 + 29 * 0 + 14, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> s29 :|: s29 >= 0, s29 <= 1 + (z' - 1) + 0 + (1 + (z'' - 1) + 0), z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s30 :|: s30 >= 0, s30 <= 1 + (z' - 1) + 0 + 0, z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s31 :|: s31 >= 0, s31 <= 1 + (z' - 1) + 0 + 0, z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 0 + (1 + (z'' - 1) + 0), z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 0 + 0, z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= 0 + 0, z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 0 + (1 + (z'' - 1) + 0), z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> s36 :|: s36 >= 0, s36 <= 0 + 0, z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s37 :|: s37 >= 0, s37 <= 0 + 0, z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s75 :|: s75 >= 0, s75 <= 29 * z1 + 29 * (1 + z0 + z'') + 29 * z' + 14, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> s78 :|: s76 >= 0, s76 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s77 >= 0, s77 <= 29 * (1 + z023 + (1 + z111 + z210)) + 29 * 0 + 29 * 0 + 14, s78 >= 0, s78 <= s76 + s77, z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s80 :|: s79 >= 0, s79 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s80 >= 0, s80 <= s79 + (1 + (z'' - 1) + 0), z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s82 :|: s81 >= 0, s81 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s82 >= 0, s82 <= s81 + 0, z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s84 :|: s83 >= 0, s83 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s84 >= 0, s84 <= s83 + 0, z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s86 :|: s85 >= 0, s85 <= 29 * (1 + z025 + (1 + z112 + z211)) + 29 * 0 + 29 * 0 + 14, s86 >= 0, s86 <= 1 + (z' - 1) + 0 + s85, z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> s88 :|: s87 >= 0, s87 <= 29 * (1 + z027 + (1 + z113 + z212)) + 29 * 0 + 29 * 0 + 14, s88 >= 0, s88 <= 0 + s87, z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> s90 :|: s89 >= 0, s89 <= 29 * (1 + z029 + (1 + z114 + z213)) + 29 * 0 + 29 * 0 + 14, s90 >= 0, s90 <= 0 + s89, z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] lteq: runtime: O(1) [0], size: O(1) [2] MERGE: runtime: O(n^1) [3 + z + 2*z'], size: O(n^2) [4 + 2*z + z*z' + 7*z' + 2*z'^2] MERGE[ITE]: runtime: O(n^1) [2 + z' + 2*z''], size: O(n^2) [10 + 4*z' + 2*z'*z'' + 14*z'' + 4*z''^2] merge[Ite]: runtime: O(1) [0], size: O(n^1) [z' + z''] merge: runtime: O(1) [0], size: O(n^1) [z + z'] splitmerge: runtime: O(1) [0], size: O(n^1) [14 + 29*z + 29*z' + 29*z''] mergesort: runtime: O(1) [0], size: O(n^1) [37*z] MERGESORT: runtime: ?, size: O(1) [0] SPLITMERGE: runtime: ?, size: INF ---------------------------------------- (83) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: MERGESORT after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (84) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 <='(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 MERGE(z, z') -{ 9 + z0' + z1 + 2*z1' + 2*z3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 14 * (1 + (1 + z1') + z3) + 2 * ((1 + (1 + z1') + z3) * (1 + (1 + z0') + z1)) + 4 * ((1 + (1 + z1') + z3) * (1 + (1 + z1') + z3)) + 4 * (1 + (1 + z0') + z1) + 10, s3 >= 0, s3 <= 2, s >= 0, s <= 1 + z1', z1 >= 0, z0' >= 0, z' = 1 + (1 + z1') + z3, z1' >= 0, z3 >= 0, z = 1 + (1 + z0') + z1 MERGE(z, z') -{ 5 + z + 2*z2 + 2*z3 }-> 1 + s7 + s' :|: s7 >= 0, s7 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + 0 + (z - 1))) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + 0 + (z - 1)) + 10, s' >= 0, s' <= z2, z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 MERGE(z, z') -{ 5 + 2*z' + z0'' + z1 }-> 1 + s8 + s'' :|: s8 >= 0, s8 <= 14 * (1 + 0 + (z' - 1)) + 2 * ((1 + 0 + (z' - 1)) * (1 + (1 + z0'') + z1)) + 4 * ((1 + 0 + (z' - 1)) * (1 + 0 + (z' - 1))) + 4 * (1 + (1 + z0'') + z1) + 10, s'' >= 0, s'' <= 0, z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0, z' - 1 >= 0 MERGE(z, z') -{ 6 + z0 + z1 + 2*z2 + 2*z3 }-> 1 + s9 + s1 :|: s9 >= 0, s9 <= 14 * (1 + z2 + z3) + 2 * ((1 + z2 + z3) * (1 + z0 + z1)) + 4 * ((1 + z2 + z3) * (1 + z2 + z3)) + 4 * (1 + z0 + z1) + 10, s1 >= 0, s1 <= z2, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 MERGE[ITE](z, z', z'') -{ 3 + z' + 2*z2 }-> 1 + s27 :|: s27 >= 0, s27 <= 4 + 2 * z' + 7 * z2 + z2 * z' + 2 * (z2 * z2), z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 3 + 2*z'' + z1 }-> 1 + s28 :|: s28 >= 0, s28 <= 4 + 2 * z1 + 7 * z'' + z'' * z1 + 2 * (z'' * z''), z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s10 + MERGESORT(1 + (z' - 1) + 0) :|: s10 >= 0, s10 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s11 + MERGESORT(1 + (z' - 1) + 0) :|: s11 >= 0, s11 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s12 + MERGESORT(1 + (z' - 1) + 0) :|: s12 >= 0, s12 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s13 + MERGESORT(0) :|: s13 >= 0, s13 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z'' - 1 >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s14 + MERGESORT(0) :|: s14 >= 0, s14 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s15 + MERGESORT(0) :|: s15 >= 0, s15 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s16 + MERGESORT(z') :|: s16 >= 0, s16 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z = 0, z'' - 1 >= 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s17 + MERGESORT(z') :|: s17 >= 0, s17 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s18 + MERGESORT(z') :|: s18 >= 0, s18 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' + 2*z'' }-> 1 + s19 + MERGESORT(1 + (z'' - 1) + 0) :|: s19 >= 0, s19 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * (1 + (z' - 1) + 0) + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' - 1 >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s20 + MERGESORT(0) :|: s20 >= 0, s20 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' = 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + z' }-> 1 + s21 + MERGESORT(z'') :|: s21 >= 0, s21 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * 0 + 0 * (1 + (z' - 1) + 0) + 2 * (0 * 0), z'' >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s22 + MERGESORT(1 + (z'' - 1) + 0) :|: s22 >= 0, s22 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z = 0, z'' - 1 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s23 + MERGESORT(z'') :|: s23 >= 0, s23 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z = 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*z'' }-> 1 + s24 + MERGESORT(1 + (z'' - 1) + 0) :|: s24 >= 0, s24 <= 4 + 2 * 0 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * 0 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z' >= 0, z'' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s25 + MERGESORT(0) :|: s25 >= 0, s25 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' = 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 }-> 1 + s26 + MERGESORT(z'') :|: s26 >= 0, s26 <= 4 + 2 * 0 + 7 * 0 + 0 * 0 + 2 * (0 * 0), z'' >= 0, z' >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + s44 + 2*s45 }-> 1 + s46 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s44 >= 0, s44 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s45 >= 0, s45 <= 29 * (1 + z03 + (1 + z11 + z2'')) + 29 * 0 + 29 * 0 + 14, s46 >= 0, s46 <= 4 + 2 * s44 + 7 * s45 + s45 * s44 + 2 * (s45 * s45), z11 >= 0, z01 >= 0, z'' = 1 + z03 + (1 + z11 + z2''), z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z03 >= 0, z = 0, z2'' >= 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + s47 + 2*z'' }-> 1 + s48 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s47 >= 0, s47 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s48 >= 0, s48 <= 4 + 2 * s47 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * s47 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z'' - 1 >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + s49 }-> 1 + s50 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s49 >= 0, s49 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s50 >= 0, s50 <= 4 + 2 * s49 + 7 * 0 + 0 * s49 + 2 * (0 * 0), z'' = 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + s51 }-> 1 + s52 + MERGESORT(1 + z01 + (1 + z1'' + z2')) :|: s51 >= 0, s51 <= 29 * (1 + z01 + (1 + z1'' + z2')) + 29 * 0 + 29 * 0 + 14, s52 >= 0, s52 <= 4 + 2 * s51 + 7 * 0 + 0 * s51 + 2 * (0 * 0), z'' >= 0, z01 >= 0, z2' >= 0, z' = 1 + z01 + (1 + z1'' + z2'), z = 0, z1'' >= 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s53 + z' }-> 1 + s54 + MERGESORT(1 + (z' - 1) + 0) :|: s53 >= 0, s53 <= 29 * (1 + z05 + (1 + z12 + z21)) + 29 * 0 + 29 * 0 + 14, s54 >= 0, s54 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * s53 + s53 * (1 + (z' - 1) + 0) + 2 * (s53 * s53), z21 >= 0, z' - 1 >= 0, z'' = 1 + z05 + (1 + z12 + z21), z12 >= 0, z05 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s55 }-> 1 + s56 + MERGESORT(0) :|: s55 >= 0, s55 <= 29 * (1 + z07 + (1 + z13 + z22)) + 29 * 0 + 29 * 0 + 14, s56 >= 0, s56 <= 4 + 2 * 0 + 7 * s55 + s55 * 0 + 2 * (s55 * s55), z07 >= 0, z'' = 1 + z07 + (1 + z13 + z22), z22 >= 0, z = 0, z13 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s57 }-> 1 + s58 + MERGESORT(z') :|: s57 >= 0, s57 <= 29 * (1 + z09 + (1 + z14 + z23)) + 29 * 0 + 29 * 0 + 14, s58 >= 0, s58 <= 4 + 2 * 0 + 7 * s57 + s57 * 0 + 2 * (s57 * s57), z'' = 1 + z09 + (1 + z14 + z23), z23 >= 0, z' >= 0, z09 >= 0, z = 0, z14 >= 0 SPLITMERGE(z, z', z'') -{ 4 + s59 + 2*s60 }-> 1 + s61 + MERGESORT(1 + z013 + (1 + z16 + z25)) :|: s59 >= 0, s59 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s60 >= 0, s60 <= 29 * (1 + z013 + (1 + z16 + z25)) + 29 * 0 + 29 * 0 + 14, s61 >= 0, s61 <= 4 + 2 * s59 + 7 * s60 + s60 * s59 + 2 * (s60 * s60), z25 >= 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z013 >= 0, z16 >= 0, z24 >= 0, z = 0, z'' = 1 + z013 + (1 + z16 + z25) SPLITMERGE(z, z', z'') -{ 4 + s62 + 2*z'' }-> 1 + s63 + MERGESORT(1 + (z'' - 1) + 0) :|: s62 >= 0, s62 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s63 >= 0, s63 <= 4 + 2 * s62 + 7 * (1 + (z'' - 1) + 0) + (1 + (z'' - 1) + 0) * s62 + 2 * ((1 + (z'' - 1) + 0) * (1 + (z'' - 1) + 0)), z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z'' - 1 >= 0, z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + s64 }-> 1 + s65 + MERGESORT(0) :|: s64 >= 0, s64 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s65 >= 0, s65 <= 4 + 2 * s64 + 7 * 0 + 0 * s64 + 2 * (0 * 0), z'' = 0, z011 >= 0, z15 >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + s66 }-> 1 + s67 + MERGESORT(z'') :|: s66 >= 0, s66 <= 29 * (1 + z011 + (1 + z15 + z24)) + 29 * 0 + 29 * 0 + 14, s67 >= 0, s67 <= 4 + 2 * s66 + 7 * 0 + 0 * s66 + 2 * (0 * 0), z011 >= 0, z15 >= 0, z'' >= 0, z' = 1 + z011 + (1 + z15 + z24), z24 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s68 + z' }-> 1 + s69 + MERGESORT(1 + z015 + (1 + z17 + z26)) :|: s68 >= 0, s68 <= 29 * (1 + z015 + (1 + z17 + z26)) + 29 * 0 + 29 * 0 + 14, s69 >= 0, s69 <= 4 + 2 * (1 + (z' - 1) + 0) + 7 * s68 + s68 * (1 + (z' - 1) + 0) + 2 * (s68 * s68), z'' = 1 + z015 + (1 + z17 + z26), z015 >= 0, z26 >= 0, z17 >= 0, z' - 1 >= 0, z = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s70 }-> 1 + s71 + MERGESORT(1 + z017 + (1 + z18 + z27)) :|: s70 >= 0, s70 <= 29 * (1 + z017 + (1 + z18 + z27)) + 29 * 0 + 29 * 0 + 14, s71 >= 0, s71 <= 4 + 2 * 0 + 7 * s70 + s70 * 0 + 2 * (s70 * s70), z18 >= 0, z'' = 1 + z017 + (1 + z18 + z27), z27 >= 0, z = 0, z017 >= 0, z' = 0 SPLITMERGE(z, z', z'') -{ 4 + 2*s72 }-> 1 + s73 + MERGESORT(1 + z019 + (1 + z19 + z28)) :|: s72 >= 0, s72 <= 29 * (1 + z019 + (1 + z19 + z28)) + 29 * 0 + 29 * 0 + 14, s73 >= 0, s73 <= 4 + 2 * 0 + 7 * s72 + s72 * 0 + 2 * (s72 * s72), z019 >= 0, z19 >= 0, z28 >= 0, z'' = 1 + z019 + (1 + z19 + z28), z' >= 0, z = 0 lteq(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> s38 :|: s38 >= 0, s38 <= 1 + (1 + z031) + z1 + (1 + (1 + z115) + z3), s5 >= 0, s5 <= 2, z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0, z' = 1 + (1 + z115) + z3, z115 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> s39 :|: s39 >= 0, s39 <= 1 + 0 + (z - 1) + (1 + z2 + z3), z - 1 >= 0, z' = 1 + z2 + z3, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> s40 :|: s40 >= 0, s40 <= 1 + (1 + z032) + z1 + (1 + 0 + (z' - 1)), z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1, z' - 1 >= 0 merge(z, z') -{ 0 }-> s41 :|: s41 >= 0, s41 <= 1 + z0 + z1 + (1 + z2 + z3), z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + s43 :|: s43 >= 0, s43 <= z1 + z'', z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + s42 :|: s42 >= 0, s42 <= z' + z2, z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0 mergesort(z) -{ 0 }-> s74 :|: s74 >= 0, s74 <= 29 * (1 + z0 + (1 + z1 + z2)) + 29 * 0 + 29 * 0 + 14, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: z >= 0 mergesort(z) -{ 0 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> s29 :|: s29 >= 0, s29 <= 1 + (z' - 1) + 0 + (1 + (z'' - 1) + 0), z' - 1 >= 0, z'' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s30 :|: s30 >= 0, s30 <= 1 + (z' - 1) + 0 + 0, z'' = 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s31 :|: s31 >= 0, s31 <= 1 + (z' - 1) + 0 + 0, z'' >= 0, z' - 1 >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 0 + (1 + (z'' - 1) + 0), z'' - 1 >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 0 + 0, z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= 0 + 0, z'' >= 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 0 + (1 + (z'' - 1) + 0), z' >= 0, z = 0, z'' - 1 >= 0 splitmerge(z, z', z'') -{ 0 }-> s36 :|: s36 >= 0, s36 <= 0 + 0, z'' = 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s37 :|: s37 >= 0, s37 <= 0 + 0, z'' >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> s75 :|: s75 >= 0, s75 <= 29 * z1 + 29 * (1 + z0 + z'') + 29 * z' + 14, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitmerge(z, z', z'') -{ 0 }-> s78 :|: s76 >= 0, s76 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s77 >= 0, s77 <= 29 * (1 + z023 + (1 + z111 + z210)) + 29 * 0 + 29 * 0 + 14, s78 >= 0, s78 <= s76 + s77, z021 >= 0, z110 >= 0, z29 >= 0, z'' = 1 + z023 + (1 + z111 + z210), z023 >= 0, z210 >= 0, z111 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s80 :|: s79 >= 0, s79 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s80 >= 0, s80 <= s79 + (1 + (z'' - 1) + 0), z021 >= 0, z110 >= 0, z29 >= 0, z'' - 1 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s82 :|: s81 >= 0, s81 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s82 >= 0, s82 <= s81 + 0, z'' = 0, z021 >= 0, z110 >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s84 :|: s83 >= 0, s83 <= 29 * (1 + z021 + (1 + z110 + z29)) + 29 * 0 + 29 * 0 + 14, s84 >= 0, s84 <= s83 + 0, z021 >= 0, z110 >= 0, z'' >= 0, z29 >= 0, z = 0, z' = 1 + z021 + (1 + z110 + z29) splitmerge(z, z', z'') -{ 0 }-> s86 :|: s85 >= 0, s85 <= 29 * (1 + z025 + (1 + z112 + z211)) + 29 * 0 + 29 * 0 + 14, s86 >= 0, s86 <= 1 + (z' - 1) + 0 + s85, z025 >= 0, z' - 1 >= 0, z112 >= 0, z211 >= 0, z'' = 1 + z025 + (1 + z112 + z211), z = 0 splitmerge(z, z', z'') -{ 0 }-> s88 :|: s87 >= 0, s87 <= 29 * (1 + z027 + (1 + z113 + z212)) + 29 * 0 + 29 * 0 + 14, s88 >= 0, s88 <= 0 + s87, z212 >= 0, z113 >= 0, z027 >= 0, z = 0, z'' = 1 + z027 + (1 + z113 + z212), z' = 0 splitmerge(z, z', z'') -{ 0 }-> s90 :|: s89 >= 0, s89 <= 29 * (1 + z029 + (1 + z114 + z213)) + 29 * 0 + 29 * 0 + 14, s90 >= 0, s90 <= 0 + s89, z029 >= 0, z'' = 1 + z029 + (1 + z114 + z213), z114 >= 0, z213 >= 0, z' >= 0, z = 0 splitmerge(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {MERGESORT,SPLITMERGE} Previous analysis results are: <=': runtime: O(1) [0], size: O(n^1) [z'] lteq: runtime: O(1) [0], size: O(1) [2] MERGE: runtime: O(n^1) [3 + z + 2*z'], size: O(n^2) [4 + 2*z + z*z' + 7*z' + 2*z'^2] MERGE[ITE]: runtime: O(n^1) [2 + z' + 2*z''], size: O(n^2) [10 + 4*z' + 2*z'*z'' + 14*z'' + 4*z''^2] merge[Ite]: runtime: O(1) [0], size: O(n^1) [z' + z''] merge: runtime: O(1) [0], size: O(n^1) [z + z'] splitmerge: runtime: O(1) [0], size: O(n^1) [14 + 29*z + 29*z' + 29*z''] mergesort: runtime: O(1) [0], size: O(n^1) [37*z] MERGESORT: runtime: INF, size: O(1) [0] SPLITMERGE: runtime: ?, size: INF ---------------------------------------- (85) 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: <='(v0, v1) -> null_<=' [0] MERGE[ITE](v0, v1, v2) -> null_MERGE[ITE] [0] lteq(v0, v1) -> null_lteq [0] mergesort(v0) -> null_mergesort [0] splitmerge(v0, v1, v2) -> null_splitmerge [0] merge(v0, v1) -> null_merge [0] merge[Ite](v0, v1, v2) -> null_merge[Ite] [0] MERGESORT(v0) -> null_MERGESORT [0] MERGE(v0, v1) -> null_MERGE [0] SPLITMERGE(v0, v1, v2) -> null_SPLITMERGE [0] And the following fresh constants: null_<=', null_MERGE[ITE], null_lteq, null_mergesort, null_splitmerge, null_merge, null_merge[Ite], null_MERGESORT, null_MERGE, null_SPLITMERGE ---------------------------------------- (86) 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: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) [1] MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](lteq(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) [1] MERGE(Cons(z0, z1), Nil) -> c9 [1] MERGE(Nil, z0) -> c10 [1] SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) [1] SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) [1] SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) [1] <='(S(z0), S(z1)) -> c(<='(z0, z1)) [0] MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) [0] MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) [0] lteq(S(z0), S(z1)) -> lteq(z0, z1) [0] lteq(0, z0) -> True [0] lteq(S(z0), 0) -> False [0] mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) [0] mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) [0] mergesort(Nil) -> Nil [0] splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) [0] splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) [0] merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](lteq(z0, z2), Cons(z0, z1), Cons(z2, z3)) [0] merge(Cons(z0, z1), Nil) -> Cons(z0, z1) [0] merge(Nil, z0) -> z0 [0] merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) [0] merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) [0] <='(v0, v1) -> null_<=' [0] MERGE[ITE](v0, v1, v2) -> null_MERGE[ITE] [0] lteq(v0, v1) -> null_lteq [0] mergesort(v0) -> null_mergesort [0] splitmerge(v0, v1, v2) -> null_splitmerge [0] merge(v0, v1) -> null_merge [0] merge[Ite](v0, v1, v2) -> null_merge[Ite] [0] MERGESORT(v0) -> null_MERGESORT [0] MERGE(v0, v1) -> null_MERGE [0] SPLITMERGE(v0, v1, v2) -> null_SPLITMERGE [0] The TRS has the following type information: MERGESORT :: Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] -> c5:null_MERGESORT Cons :: S:0 -> Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] -> Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] c5 :: c11:c12:c13:null_SPLITMERGE -> c5:null_MERGESORT SPLITMERGE :: Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] -> Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] -> Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] -> c11:c12:c13:null_SPLITMERGE Nil :: Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] MERGE :: Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] -> Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] -> c8:c9:c10:null_MERGE c8 :: c3:c4:null_MERGE[ITE] -> c:null_<=' -> c8:c9:c10:null_MERGE MERGE[ITE] :: False:True:null_lteq -> Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] -> Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] -> c3:c4:null_MERGE[ITE] lteq :: S:0 -> S:0 -> False:True:null_lteq <=' :: S:0 -> S:0 -> c:null_<=' c9 :: c8:c9:c10:null_MERGE c10 :: c8:c9:c10:null_MERGE c11 :: c11:c12:c13:null_SPLITMERGE -> c11:c12:c13:null_SPLITMERGE c12 :: c8:c9:c10:null_MERGE -> c5:null_MERGESORT -> c11:c12:c13:null_SPLITMERGE mergesort :: Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] -> Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] c13 :: c8:c9:c10:null_MERGE -> c5:null_MERGESORT -> c11:c12:c13:null_SPLITMERGE S :: S:0 -> S:0 c :: c:null_<=' -> c:null_<=' False :: False:True:null_lteq c3 :: c8:c9:c10:null_MERGE -> c3:c4:null_MERGE[ITE] True :: False:True:null_lteq c4 :: c8:c9:c10:null_MERGE -> c3:c4:null_MERGE[ITE] 0 :: S:0 splitmerge :: Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] -> Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] -> Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] -> Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] merge :: Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] -> Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] -> Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] merge[Ite] :: False:True:null_lteq -> Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] -> Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] -> Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] null_<=' :: c:null_<=' null_MERGE[ITE] :: c3:c4:null_MERGE[ITE] null_lteq :: False:True:null_lteq null_mergesort :: Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] null_splitmerge :: Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] null_merge :: Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] null_merge[Ite] :: Cons:Nil:null_mergesort:null_splitmerge:null_merge:null_merge[Ite] null_MERGESORT :: c5:null_MERGESORT null_MERGE :: c8:c9:c10:null_MERGE null_SPLITMERGE :: c11:c12:c13:null_SPLITMERGE Rewrite Strategy: INNERMOST ---------------------------------------- (87) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: Nil => 0 c9 => 1 c10 => 0 False => 1 True => 2 0 => 0 null_<=' => 0 null_MERGE[ITE] => 0 null_lteq => 0 null_mergesort => 0 null_splitmerge => 0 null_merge => 0 null_merge[Ite] => 0 null_MERGESORT => 0 null_MERGE => 0 null_SPLITMERGE => 0 ---------------------------------------- (88) Obligation: Complexity RNTS consisting of the following rules: <='(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 <='(z, z') -{ 0 }-> 1 + <='(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 MERGE(z, z') -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 MERGE(z, z') -{ 1 }-> 0 :|: z0 >= 0, z = 0, z' = z0 MERGE(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 MERGE(z, z') -{ 1 }-> 1 + MERGE[ITE](lteq(z0, z2), 1 + z0 + z1, 1 + z2 + z3) + <='(z0, z2) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 MERGESORT(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 MERGESORT(z) -{ 1 }-> 1 + SPLITMERGE(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z0, z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0 MERGE[ITE](z, z', z'') -{ 0 }-> 1 + MERGE(z1, z2) :|: z = 2, z'' = z2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z2 >= 0 SPLITMERGE(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + SPLITMERGE(z1, 1 + z0 + z3, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(mergesort(z0), mergesort(z1)) + MERGESORT(z0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1 SPLITMERGE(z, z', z'') -{ 1 }-> 1 + MERGE(mergesort(z0), mergesort(z1)) + MERGESORT(z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1 lteq(z, z') -{ 0 }-> lteq(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 lteq(z, z') -{ 0 }-> 2 :|: z0 >= 0, z = 0, z' = z0 lteq(z, z') -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 merge(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0 merge(z, z') -{ 0 }-> merge[Ite](lteq(z0, z2), 1 + z0 + z1, 1 + z2 + z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 merge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 merge(z, z') -{ 0 }-> 1 + z0 + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z0 + merge(z1, z2) :|: z = 2, z'' = z2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z2 >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + z1 + merge(z0, z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0 mergesort(z) -{ 0 }-> splitmerge(1 + z0 + (1 + z1 + z2), 0, 0) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0 mergesort(z) -{ 0 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 mergesort(z) -{ 0 }-> 1 + z0 + 0 :|: z0 >= 0, z = 1 + z0 + 0 splitmerge(z, z', z'') -{ 0 }-> splitmerge(z1, 1 + z0 + z3, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 splitmerge(z, z', z'') -{ 0 }-> merge(mergesort(z0), mergesort(z1)) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1 splitmerge(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (89) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) We considered the (Usable) Rules:none And the Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(<=(x_1, x_2)) = 0 POL(<='(x_1, x_2)) = 0 POL(Cons(x_1, x_2)) = [1] + x_2 POL(False) = 0 POL(MERGE(x_1, x_2)) = 0 POL(MERGESORT(x_1)) = [2]x_1^2 POL(MERGE[ITE](x_1, x_2, x_3)) = 0 POL(Nil) = 0 POL(S(x_1)) = 0 POL(SPLITMERGE(x_1, x_2, x_3)) = [2]x_1 + [2]x_3 + [2]x_3^2 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + [2]x_2^2 POL(True) = 0 POL(c(x_1)) = x_1 POL(c10) = 0 POL(c11(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9) = 0 POL(merge(x_1, x_2)) = [1] + x_1 + x_2 + x_2^2 + x_1*x_2 + x_1^2 POL(merge[Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 + x_3^2 + x_2*x_3 + x_2^2 POL(mergesort(x_1)) = 0 POL(splitmerge(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 + x_3^2 + x_2*x_3 + x_1*x_3 + x_1^2 + x_1*x_2 + x_2^2 ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) S tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3 Compound Symbols: c_1, c3_1, c4_1, c5_1, c8_2, c9, c10, c11_1, c12_2, c13_2 ---------------------------------------- (91) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) S tuples: MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3 Compound Symbols: c_1, c3_1, c4_1, c5_1, c8_2, c9, c10, c11_1, c12_2, c13_2 ---------------------------------------- (93) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 We considered the (Usable) Rules:none And the Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(<=(x_1, x_2)) = 0 POL(<='(x_1, x_2)) = 0 POL(Cons(x_1, x_2)) = [1] + x_2 POL(False) = 0 POL(MERGE(x_1, x_2)) = [1] POL(MERGESORT(x_1)) = [2]x_1^2 POL(MERGE[ITE](x_1, x_2, x_3)) = [1] POL(Nil) = 0 POL(S(x_1)) = 0 POL(SPLITMERGE(x_1, x_2, x_3)) = [1] + x_1 + [2]x_3 + [2]x_3^2 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + [2]x_2^2 POL(True) = 0 POL(c(x_1)) = x_1 POL(c10) = 0 POL(c11(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9) = 0 POL(merge(x_1, x_2)) = [1] + x_1 + x_2 + x_2^2 + x_1*x_2 + x_1^2 POL(merge[Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 + x_3^2 + x_2*x_3 + x_2^2 POL(mergesort(x_1)) = 0 POL(splitmerge(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 + x_3^2 + x_2*x_3 + x_1*x_3 + x_1^2 + x_1*x_2 + x_2^2 ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) S tuples: MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3 Compound Symbols: c_1, c3_1, c4_1, c5_1, c8_2, c9, c10, c11_1, c12_2, c13_2 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) by MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)), <='(0, z0)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)), <='(S(z0), 0)) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)), <='(0, z0)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)), <='(S(z0), 0)) S tuples: MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)), <='(0, z0)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)), <='(S(z0), 0)) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3 Compound Symbols: c_1, c3_1, c4_1, c5_1, c9, c10, c11_1, c12_2, c13_2, c8_2 ---------------------------------------- (97) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MERGE(Cons(z0, z1), Nil) -> c9 MERGE(Nil, z0) -> c10 ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)), <='(0, z0)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)), <='(S(z0), 0)) S tuples: MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)), <='(0, z0)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)), <='(S(z0), 0)) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c12_2, c13_2, c8_2 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) S tuples: MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c12_2, c13_2, c8_2, c8_1 ---------------------------------------- (101) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) We considered the (Usable) Rules: splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) <=(S(z0), 0) -> False merge(Nil, z0) -> z0 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Nil) -> Nil <=(0, z0) -> True <=(S(z0), S(z1)) -> <=(z0, z1) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) And the Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(<=(x_1, x_2)) = [1] POL(<='(x_1, x_2)) = 0 POL(Cons(x_1, x_2)) = [1] + x_2 POL(False) = 0 POL(MERGE(x_1, x_2)) = [1] + [2]x_2 POL(MERGESORT(x_1)) = [2]x_1^2 POL(MERGE[ITE](x_1, x_2, x_3)) = [2]x_3 + x_1^2 POL(Nil) = 0 POL(S(x_1)) = 0 POL(SPLITMERGE(x_1, x_2, x_3)) = [1] + x_1 + [2]x_3 + [2]x_3^2 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + [2]x_2^2 POL(True) = [1] POL(c(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(merge(x_1, x_2)) = x_1 + x_2 POL(merge[Ite](x_1, x_2, x_3)) = x_2 + x_3 POL(mergesort(x_1)) = x_1 POL(splitmerge(x_1, x_2, x_3)) = x_1 + x_2 + x_3 ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) S tuples: MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c12_2, c13_2, c8_2, c8_1 ---------------------------------------- (103) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) We considered the (Usable) Rules: splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) <=(S(z0), 0) -> False merge(Nil, z0) -> z0 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Nil) -> Nil <=(0, z0) -> True <=(S(z0), S(z1)) -> <=(z0, z1) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) And the Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(<=(x_1, x_2)) = [2] POL(<='(x_1, x_2)) = 0 POL(Cons(x_1, x_2)) = [2] + x_2 POL(False) = [2] POL(MERGE(x_1, x_2)) = [2]x_2 + x_1*x_2 POL(MERGESORT(x_1)) = x_1^2 POL(MERGE[ITE](x_1, x_2, x_3)) = x_2*x_3 + x_1*x_3 POL(Nil) = 0 POL(S(x_1)) = 0 POL(SPLITMERGE(x_1, x_2, x_3)) = [2]x_3 + x_3^2 + x_2*x_3 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + x_2^2 POL(True) = 0 POL(c(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(merge(x_1, x_2)) = x_1 + x_2 POL(merge[Ite](x_1, x_2, x_3)) = x_2 + x_3 POL(mergesort(x_1)) = x_1 POL(splitmerge(x_1, x_2, x_3)) = x_1 + x_2 + x_3 ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) S tuples: MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c12_2, c13_2, c8_2, c8_1 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) by SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, x0, Nil) -> c12(MERGE(mergesort(x0), Nil), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(Cons(z0, Nil))) SPLITMERGE(Nil, Nil, x1) -> c12(MERGE(Nil, mergesort(x1)), MERGESORT(Nil)) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, x0, Nil) -> c12(MERGE(mergesort(x0), Nil), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(Cons(z0, Nil))) SPLITMERGE(Nil, Nil, x1) -> c12(MERGE(Nil, mergesort(x1)), MERGESORT(Nil)) S tuples: MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c13_2, c8_2, c8_1, c12_2 ---------------------------------------- (107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SPLITMERGE(Nil, Nil, x1) -> c12(MERGE(Nil, mergesort(x1)), MERGESORT(Nil)) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, x0, Nil) -> c12(MERGE(mergesort(x0), Nil), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(Cons(z0, Nil))) S tuples: MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c13_2, c8_2, c8_1, c12_2 ---------------------------------------- (109) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) S tuples: MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c13_2, c8_2, c8_1, c12_2, c12_1 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) by SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil))) SPLITMERGE(Nil, x0, Nil) -> c13(MERGE(mergesort(x0), Nil), MERGESORT(Nil)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Nil, x1) -> c13(MERGE(Nil, mergesort(x1)), MERGESORT(x1)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil))) SPLITMERGE(Nil, x0, Nil) -> c13(MERGE(mergesort(x0), Nil), MERGESORT(Nil)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Nil, x1) -> c13(MERGE(Nil, mergesort(x1)), MERGESORT(x1)) S tuples: MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_2, c8_1, c12_2, c12_1, c13_2 ---------------------------------------- (113) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SPLITMERGE(Nil, x0, Nil) -> c13(MERGE(mergesort(x0), Nil), MERGESORT(Nil)) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Nil, x1) -> c13(MERGE(Nil, mergesort(x1)), MERGESORT(x1)) S tuples: MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_2, c8_1, c12_2, c12_1, c13_2 ---------------------------------------- (115) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) SPLITMERGE(Nil, Nil, x1) -> c13(MERGESORT(x1)) S tuples: MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_2, c8_1, c12_2, c12_1, c13_2, c13_1 ---------------------------------------- (117) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: SPLITMERGE(Nil, Nil, x1) -> c13(MERGESORT(x1)) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) S tuples: MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_2, c8_1, c12_2, c12_1, c13_2, c13_1 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) by MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (121) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) We considered the (Usable) Rules: splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Nil) -> Nil mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) merge(Nil, z0) -> z0 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) And the Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(<=(x_1, x_2)) = 0 POL(<='(x_1, x_2)) = 0 POL(Cons(x_1, x_2)) = [2] + x_2 POL(False) = 0 POL(MERGE(x_1, x_2)) = x_1*x_2 POL(MERGESORT(x_1)) = x_1^2 POL(MERGE[ITE](x_1, x_2, x_3)) = x_2*x_3 POL(Nil) = 0 POL(S(x_1)) = 0 POL(SPLITMERGE(x_1, x_2, x_3)) = x_3^2 + x_2*x_3 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + x_2^2 POL(True) = 0 POL(c(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(merge(x_1, x_2)) = x_1 + x_2 POL(merge[Ite](x_1, x_2, x_3)) = x_2 + x_3 POL(mergesort(x_1)) = x_1 POL(splitmerge(x_1, x_2, x_3)) = x_1 + x_2 + x_3 ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (123) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) We considered the (Usable) Rules: splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) <=(S(z0), 0) -> False merge(Nil, z0) -> z0 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Nil) -> Nil <=(0, z0) -> True <=(S(z0), S(z1)) -> <=(z0, z1) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) And the Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(<=(x_1, x_2)) = [1] POL(<='(x_1, x_2)) = 0 POL(Cons(x_1, x_2)) = [2] + x_2 POL(False) = [1] POL(MERGE(x_1, x_2)) = x_2 + x_1*x_2 POL(MERGESORT(x_1)) = x_1^2 POL(MERGE[ITE](x_1, x_2, x_3)) = x_2*x_3 + x_1*x_3 POL(Nil) = 0 POL(S(x_1)) = 0 POL(SPLITMERGE(x_1, x_2, x_3)) = x_3 + x_3^2 + x_2*x_3 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + x_2^2 POL(True) = 0 POL(c(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(merge(x_1, x_2)) = x_1 + x_2 POL(merge[Ite](x_1, x_2, x_3)) = x_2 + x_3 POL(mergesort(x_1)) = x_1 POL(splitmerge(x_1, x_2, x_3)) = x_1 + x_2 + x_3 ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (125) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) We considered the (Usable) Rules: splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) <=(S(z0), 0) -> False merge(Nil, z0) -> z0 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Nil) -> Nil <=(0, z0) -> True <=(S(z0), S(z1)) -> <=(z0, z1) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) And the Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(<=(x_1, x_2)) = [1] POL(<='(x_1, x_2)) = 0 POL(Cons(x_1, x_2)) = [1] + x_2 POL(False) = 0 POL(MERGE(x_1, x_2)) = [2] + [2]x_2 POL(MERGESORT(x_1)) = [2]x_1^2 POL(MERGE[ITE](x_1, x_2, x_3)) = [2]x_3 + [2]x_1^2 POL(Nil) = 0 POL(S(x_1)) = 0 POL(SPLITMERGE(x_1, x_2, x_3)) = [2] + x_1 + [2]x_3 + [2]x_3^2 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + [2]x_2^2 POL(True) = [1] POL(c(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(merge(x_1, x_2)) = x_1 + x_2 POL(merge[Ite](x_1, x_2, x_3)) = x_2 + x_3 POL(mergesort(x_1)) = x_1 POL(splitmerge(x_1, x_2, x_3)) = x_1 + x_2 + x_3 ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0)) by SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Nil))) SPLITMERGE(Nil, Nil, Cons(x1, Cons(x2, x3))) -> c12(MERGE(Nil, splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Nil)) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Nil))) SPLITMERGE(Nil, Nil, Cons(x1, Cons(x2, x3))) -> c12(MERGE(Nil, splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Nil)) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SPLITMERGE(Nil, Nil, Cons(x1, Cons(x2, x3))) -> c12(MERGE(Nil, splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Nil)) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Nil))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (131) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Nil))) SPLITMERGE(Nil, Nil, Cons(x1, Nil)) -> c12(MERGE(Nil, Cons(x1, Nil)), MERGESORT(Nil)) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Nil))) SPLITMERGE(Nil, Nil, Cons(x1, Nil)) -> c12(MERGE(Nil, Cons(x1, Nil)), MERGESORT(Nil)) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (135) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SPLITMERGE(Nil, Nil, Cons(x1, Nil)) -> c12(MERGE(Nil, Cons(x1, Nil)), MERGESORT(Nil)) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Nil))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (137) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (139) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) by SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Cons(z1, z2))) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Nil), MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Nil), MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2 ---------------------------------------- (141) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) by SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Cons(z1, z2))) -> c12(MERGE(Cons(x0, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c12(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Nil) -> c12(MERGE(Cons(x0, Nil), Nil)) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Nil), Nil) -> c12(MERGE(Cons(x0, Nil), Nil)) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2 ---------------------------------------- (145) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SPLITMERGE(Nil, Cons(x0, Nil), Nil) -> c12(MERGE(Cons(x0, Nil), Nil)) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2 ---------------------------------------- (147) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) by SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Nil, Cons(x1, Cons(x2, x3))) -> c13(MERGE(Nil, splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Nil, Cons(x1, Cons(x2, x3))) -> c13(MERGE(Nil, splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2 ---------------------------------------- (149) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Nil, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2 ---------------------------------------- (151) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: SPLITMERGE(Nil, Nil, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2 ---------------------------------------- (153) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) by SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Cons(z1, z2))) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Nil), MERGESORT(Nil)) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Nil), MERGESORT(Nil)) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2 ---------------------------------------- (155) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Nil), MERGESORT(Nil)) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2 ---------------------------------------- (157) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2 ---------------------------------------- (159) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) by SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Cons(z1, z2))) -> c13(MERGE(Cons(x0, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Nil) -> c13(MERGE(Cons(x0, Nil), Nil), MERGESORT(Nil)) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Nil) -> c13(MERGE(Cons(x0, Nil), Nil), MERGESORT(Nil)) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_1, c8_2, c12_2, c13_2 ---------------------------------------- (161) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SPLITMERGE(Nil, Cons(x0, Nil), Nil) -> c13(MERGE(Cons(x0, Nil), Nil), MERGESORT(Nil)) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_1, c8_2, c12_2, c13_2 ---------------------------------------- (163) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_1, c8_2, c12_2, c13_2 ---------------------------------------- (165) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c13(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Nil, Cons(x1, Nil)) -> c13(MERGE(Nil, Cons(x1, Nil))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Nil, Cons(x1, Nil)) -> c13(MERGE(Nil, Cons(x1, Nil))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1 ---------------------------------------- (167) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SPLITMERGE(Nil, Nil, Cons(x1, Nil)) -> c13(MERGE(Nil, Cons(x1, Nil))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1 ---------------------------------------- (169) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) by SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1 ---------------------------------------- (171) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: <='(S(z0), S(z1)) -> c(<='(z0, z1)) MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1 ---------------------------------------- (173) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace <='(S(z0), S(z1)) -> c(<='(z0, z1)) by <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2, <='_2 Compound Symbols: c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1, c_1 ---------------------------------------- (175) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2, <='_2 Compound Symbols: c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1, c_1 ---------------------------------------- (177) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) by SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2, <='_2 Compound Symbols: c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1, c_1 ---------------------------------------- (179) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2, <='_2 Compound Symbols: c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1, c_1 ---------------------------------------- (181) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Cons(z1, z2))) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2, <='_2 Compound Symbols: c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1, c_1 ---------------------------------------- (183) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) by MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2, <='_2 Compound Symbols: c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1, c_1, c3_1 ---------------------------------------- (185) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) by MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, SPLITMERGE_3, MERGE_2, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1, c_1, c3_1, c4_1 ---------------------------------------- (187) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, SPLITMERGE_3, MERGE_2, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1, c_1, c3_1, c4_1 ---------------------------------------- (189) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, SPLITMERGE_3, MERGE_2, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c11_1, c8_1, c12_1, c8_2, c13_2, c13_1, c12_2, c_1, c3_1, c4_1 ---------------------------------------- (191) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) by SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c12_1, c8_2, c13_2, c13_1, c12_2, c_1, c3_1, c4_1, c11_1 ---------------------------------------- (193) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Cons(z1, z2))) -> c12(MERGE(Cons(x0, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c12_2, c_1, c3_1, c4_1, c11_1 ---------------------------------------- (195) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Cons(z1, z2))) -> c12(MERGE(Cons(x0, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil))) by SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c12_2, c_1, c3_1, c4_1, c11_1 ---------------------------------------- (197) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) by SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c12_2, c_1, c3_1, c4_1, c11_1 ---------------------------------------- (199) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) by MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c12_2, c_1, c3_1, c4_1, c11_1 ---------------------------------------- (201) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) by SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c12_2, c_1, c3_1, c4_1, c11_1 ---------------------------------------- (203) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c12_2, c_1, c3_1, c4_1, c11_1 ---------------------------------------- (205) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) by SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c12_2, c_1, c3_1, c4_1, c11_1 ---------------------------------------- (207) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) by SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3)))) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c12_2, c_1, c3_1, c4_1, c11_1 ---------------------------------------- (209) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) by SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3))) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c12_2, c_1, c3_1, c4_1, c11_1 ---------------------------------------- (211) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Cons(z1, z2))) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c12_2, c_1, c3_1, c4_1, c11_1 ---------------------------------------- (213) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil))) ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c12_2, c_1, c3_1, c4_1, c11_1 ---------------------------------------- (215) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Cons(z1, z2))) -> c13(MERGE(Cons(x0, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) by SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_1, c12_2, c_1, c3_1, c4_1, c11_1, c13_2 ---------------------------------------- (217) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil))) ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_1, c12_2, c_1, c3_1, c4_1, c11_1, c13_2 ---------------------------------------- (219) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) by SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4))) ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_1, c12_2, c_1, c3_1, c4_1, c11_1, c13_2 ---------------------------------------- (221) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2)))) ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_1, c12_2, c_1, c3_1, c4_1, c11_1, c13_2 ---------------------------------------- (223) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) by SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil)))) ---------------------------------------- (224) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_1, c12_2, c_1, c3_1, c4_1, c11_1, c13_2 ---------------------------------------- (225) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) ---------------------------------------- (226) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_1, c12_2, c_1, c3_1, c4_1, c11_1, c13_2 ---------------------------------------- (227) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2)))) ---------------------------------------- (228) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_1, c_1, c12_2, c3_1, c4_1, c11_1, c13_2 ---------------------------------------- (229) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2)))) by SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(x0, Nil)), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2)))) ---------------------------------------- (230) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(x0, Nil)), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_1, c_1, c12_2, c3_1, c4_1, c11_1, c13_2 ---------------------------------------- (231) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) by SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) ---------------------------------------- (232) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(x0, Nil)), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2)))) SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_1, c_1, c12_2, c3_1, c4_1, c11_1, c13_2 ---------------------------------------- (233) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (234) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(x0, Nil)), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2)))) SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c1(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c1(MERGESORT(Cons(z1, Cons(z2, z3)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_1, c_1, c12_2, c3_1, c4_1, c11_1, c13_2, c1_1 ---------------------------------------- (235) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c1(MERGESORT(Cons(z1, Cons(z2, z3)))) ---------------------------------------- (236) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(x0, Nil)), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2)))) SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c1(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_1, c_1, c3_1, c4_1, c12_2, c11_1, c13_2, c1_1 ---------------------------------------- (237) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3)))) by SPLITMERGE(Nil, Cons(x0, Cons(x4, x5)), Cons(x2, Cons(z3, z4))) -> c13(MERGE(mergesort(Cons(x0, Cons(x4, x5))), splitmerge(Cons(z3, z4), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z3, z4)))) ---------------------------------------- (238) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(x0, Nil)), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2)))) SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c1(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(x0, Cons(x4, x5)), Cons(x2, Cons(z3, z4))) -> c13(MERGE(mergesort(Cons(x0, Cons(x4, x5))), splitmerge(Cons(z3, z4), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z3, z4)))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_1, c_1, c3_1, c4_1, c12_2, c11_1, c13_2, c1_1 ---------------------------------------- (239) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) by SPLITMERGE(Nil, Cons(0, Nil), Cons(z1, Nil)) -> c12(MERGE(Cons(0, Nil), Cons(z1, Nil))) SPLITMERGE(Nil, Cons(S(y0), Nil), Cons(0, Nil)) -> c12(MERGE(Cons(S(y0), Nil), Cons(0, Nil))) SPLITMERGE(Nil, Cons(S(S(y0)), Nil), Cons(S(S(y2)), Nil)) -> c12(MERGE(Cons(S(S(y0)), Nil), Cons(S(S(y2)), Nil))) SPLITMERGE(Nil, Cons(S(0), Nil), Cons(S(y1), Nil)) -> c12(MERGE(Cons(S(0), Nil), Cons(S(y1), Nil))) SPLITMERGE(Nil, Cons(S(S(y0)), Nil), Cons(S(0), Nil)) -> c12(MERGE(Cons(S(S(y0)), Nil), Cons(S(0), Nil))) ---------------------------------------- (240) Obligation: Complexity Dependency Tuples Problem Rules: <=(S(z0), S(z1)) -> <=(z0, z1) <=(0, z0) -> True <=(S(z0), 0) -> False mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) mergesort(Nil) -> Nil splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) merge(Cons(z0, z1), Nil) -> Cons(z0, z1) merge(Nil, z0) -> z0 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) Tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil))) <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z3, Cons(z4, z5)))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil))) SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil)))) SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(x0, Nil)), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2)))) SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c1(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil))) SPLITMERGE(Nil, Cons(x0, Cons(x4, x5)), Cons(x2, Cons(z3, z4))) -> c13(MERGE(mergesort(Cons(x0, Cons(x4, x5))), splitmerge(Cons(z3, z4), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z3, z4)))) SPLITMERGE(Nil, Cons(0, Nil), Cons(z1, Nil)) -> c12(MERGE(Cons(0, Nil), Cons(z1, Nil))) SPLITMERGE(Nil, Cons(S(y0), Nil), Cons(0, Nil)) -> c12(MERGE(Cons(S(y0), Nil), Cons(0, Nil))) SPLITMERGE(Nil, Cons(S(S(y0)), Nil), Cons(S(S(y2)), Nil)) -> c12(MERGE(Cons(S(S(y0)), Nil), Cons(S(S(y2)), Nil))) SPLITMERGE(Nil, Cons(S(0), Nil), Cons(S(y1), Nil)) -> c12(MERGE(Cons(S(0), Nil), Cons(S(y1), Nil))) SPLITMERGE(Nil, Cons(S(S(y0)), Nil), Cons(S(0), Nil)) -> c12(MERGE(Cons(S(S(y0)), Nil), Cons(S(0), Nil))) S tuples: MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) K tuples: MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, <='_2, MERGE[ITE]_3 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c13_1, c_1, c3_1, c4_1, c12_2, c11_1, c13_2, c1_1 ---------------------------------------- (241) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (242) 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: mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil) [1] mergesort(Cons(x, Nil)) -> Cons(x, Nil) [1] merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs)) [1] merge(Cons(x, xs), Nil) -> Cons(x, xs) [1] splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1) [1] splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2)) [1] mergesort(Nil) -> Nil [1] merge(Nil, xs2) -> xs2 [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] goal(xs) -> mergesort(xs) [1] <=(S(x), S(y)) -> <=(x, y) [0] <=(0, y) -> True [0] <=(S(x), 0) -> False [0] merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs)) [0] merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2)) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (243) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: <= => lteq ---------------------------------------- (244) 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: mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil) [1] mergesort(Cons(x, Nil)) -> Cons(x, Nil) [1] merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](lteq(x', x), Cons(x', xs'), Cons(x, xs)) [1] merge(Cons(x, xs), Nil) -> Cons(x, xs) [1] splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1) [1] splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2)) [1] mergesort(Nil) -> Nil [1] merge(Nil, xs2) -> xs2 [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] goal(xs) -> mergesort(xs) [1] lteq(S(x), S(y)) -> lteq(x, y) [0] lteq(0, y) -> True [0] lteq(S(x), 0) -> False [0] merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs)) [0] merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2)) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (245) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (246) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil) [1] mergesort(Cons(x, Nil)) -> Cons(x, Nil) [1] merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](lteq(x', x), Cons(x', xs'), Cons(x, xs)) [1] merge(Cons(x, xs), Nil) -> Cons(x, xs) [1] splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1) [1] splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2)) [1] mergesort(Nil) -> Nil [1] merge(Nil, xs2) -> xs2 [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] goal(xs) -> mergesort(xs) [1] lteq(S(x), S(y)) -> lteq(x, y) [0] lteq(0, y) -> True [0] lteq(S(x), 0) -> False [0] merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs)) [0] merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2)) [0] The TRS has the following type information: mergesort :: Cons:Nil -> Cons:Nil Cons :: S:0 -> Cons:Nil -> Cons:Nil splitmerge :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil merge :: Cons:Nil -> Cons:Nil -> Cons:Nil merge[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil lteq :: S:0 -> S:0 -> True:False notEmpty :: Cons:Nil -> True:False True :: True:False False :: True:False goal :: Cons:Nil -> Cons:Nil S :: S:0 -> S:0 0 :: S:0 Rewrite Strategy: INNERMOST ---------------------------------------- (247) 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: notEmpty_1 goal_1 (c) The following functions are completely defined: mergesort_1 splitmerge_3 merge_2 lteq_2 merge[Ite]_3 Due to the following rules being added: lteq(v0, v1) -> null_lteq [0] merge[Ite](v0, v1, v2) -> Nil [0] And the following fresh constants: null_lteq ---------------------------------------- (248) 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: mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil) [1] mergesort(Cons(x, Nil)) -> Cons(x, Nil) [1] merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](lteq(x', x), Cons(x', xs'), Cons(x, xs)) [1] merge(Cons(x, xs), Nil) -> Cons(x, xs) [1] splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1) [1] splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2)) [1] mergesort(Nil) -> Nil [1] merge(Nil, xs2) -> xs2 [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] goal(xs) -> mergesort(xs) [1] lteq(S(x), S(y)) -> lteq(x, y) [0] lteq(0, y) -> True [0] lteq(S(x), 0) -> False [0] merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs)) [0] merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2)) [0] lteq(v0, v1) -> null_lteq [0] merge[Ite](v0, v1, v2) -> Nil [0] The TRS has the following type information: mergesort :: Cons:Nil -> Cons:Nil Cons :: S:0 -> Cons:Nil -> Cons:Nil splitmerge :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil merge :: Cons:Nil -> Cons:Nil -> Cons:Nil merge[Ite] :: True:False:null_lteq -> Cons:Nil -> Cons:Nil -> Cons:Nil lteq :: S:0 -> S:0 -> True:False:null_lteq notEmpty :: Cons:Nil -> True:False:null_lteq True :: True:False:null_lteq False :: True:False:null_lteq goal :: Cons:Nil -> Cons:Nil S :: S:0 -> S:0 0 :: S:0 null_lteq :: True:False:null_lteq Rewrite Strategy: INNERMOST ---------------------------------------- (249) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (250) 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: mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil) [1] mergesort(Cons(x, Nil)) -> Cons(x, Nil) [1] merge(Cons(S(x''), xs'), Cons(S(y'), xs)) -> merge[Ite](lteq(x'', y'), Cons(S(x''), xs'), Cons(S(y'), xs)) [1] merge(Cons(0, xs'), Cons(x, xs)) -> merge[Ite](True, Cons(0, xs'), Cons(x, xs)) [1] merge(Cons(S(x1), xs'), Cons(0, xs)) -> merge[Ite](False, Cons(S(x1), xs'), Cons(0, xs)) [1] merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](null_lteq, Cons(x', xs'), Cons(x, xs)) [1] merge(Cons(x, xs), Nil) -> Cons(x, xs) [1] splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1) [1] splitmerge(Nil, Cons(x''', Cons(x2, xs'')), Cons(x'1, Cons(x4, xs3))) -> merge(splitmerge(Cons(x''', Cons(x2, xs'')), Nil, Nil), splitmerge(Cons(x'1, Cons(x4, xs3)), Nil, Nil)) [3] splitmerge(Nil, Cons(x''', Cons(x2, xs'')), Cons(x5, Nil)) -> merge(splitmerge(Cons(x''', Cons(x2, xs'')), Nil, Nil), Cons(x5, Nil)) [3] splitmerge(Nil, Cons(x''', Cons(x2, xs'')), Nil) -> merge(splitmerge(Cons(x''', Cons(x2, xs'')), Nil, Nil), Nil) [3] splitmerge(Nil, Cons(x3, Nil), Cons(x'2, Cons(x6, xs4))) -> merge(Cons(x3, Nil), splitmerge(Cons(x'2, Cons(x6, xs4)), Nil, Nil)) [3] splitmerge(Nil, Cons(x3, Nil), Cons(x7, Nil)) -> merge(Cons(x3, Nil), Cons(x7, Nil)) [3] splitmerge(Nil, Cons(x3, Nil), Nil) -> merge(Cons(x3, Nil), Nil) [3] splitmerge(Nil, Nil, Cons(x'3, Cons(x8, xs5))) -> merge(Nil, splitmerge(Cons(x'3, Cons(x8, xs5)), Nil, Nil)) [3] splitmerge(Nil, Nil, Cons(x9, Nil)) -> merge(Nil, Cons(x9, Nil)) [3] splitmerge(Nil, Nil, Nil) -> merge(Nil, Nil) [3] mergesort(Nil) -> Nil [1] merge(Nil, xs2) -> xs2 [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] goal(xs) -> mergesort(xs) [1] lteq(S(x), S(y)) -> lteq(x, y) [0] lteq(0, y) -> True [0] lteq(S(x), 0) -> False [0] merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs)) [0] merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2)) [0] lteq(v0, v1) -> null_lteq [0] merge[Ite](v0, v1, v2) -> Nil [0] The TRS has the following type information: mergesort :: Cons:Nil -> Cons:Nil Cons :: S:0 -> Cons:Nil -> Cons:Nil splitmerge :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil merge :: Cons:Nil -> Cons:Nil -> Cons:Nil merge[Ite] :: True:False:null_lteq -> Cons:Nil -> Cons:Nil -> Cons:Nil lteq :: S:0 -> S:0 -> True:False:null_lteq notEmpty :: Cons:Nil -> True:False:null_lteq True :: True:False:null_lteq False :: True:False:null_lteq goal :: Cons:Nil -> Cons:Nil S :: S:0 -> S:0 0 :: S:0 null_lteq :: True:False:null_lteq Rewrite Strategy: INNERMOST ---------------------------------------- (251) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: Nil => 0 True => 2 False => 1 0 => 0 null_lteq => 0 ---------------------------------------- (252) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> mergesort(xs) :|: xs >= 0, z = xs lteq(z, z') -{ 0 }-> lteq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x lteq(z, z') -{ 0 }-> 2 :|: y >= 0, z = 0, z' = y lteq(z, z') -{ 0 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 merge(z, z') -{ 1 }-> xs2 :|: xs2 >= 0, z' = xs2, z = 0 merge(z, z') -{ 1 }-> merge[Ite](lteq(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0 merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z = 1 + 0 + xs', xs' >= 0, x >= 0 merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + xs) :|: xs >= 0, x1 >= 0, z' = 1 + 0 + xs, z = 1 + (1 + x1) + xs', xs' >= 0 merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, xs2) :|: z = 2, xs >= 0, xs2 >= 0, z' = 1 + x + xs, x >= 0, z'' = xs2 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs1, xs) :|: xs >= 0, z = 1, x >= 0, xs1 >= 0, z'' = 1 + x + xs, z' = xs1 mergesort(z) -{ 1 }-> splitmerge(1 + x' + (1 + x + xs), 0, 0) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) mergesort(z) -{ 1 }-> 0 :|: z = 0 mergesort(z) -{ 1 }-> 1 + x + 0 :|: x >= 0, z = 1 + x + 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 splitmerge(z, z', z'') -{ 1 }-> splitmerge(xs, 1 + x + xs2, xs1) :|: z = 1 + x + xs, xs >= 0, xs2 >= 0, x >= 0, xs1 >= 0, z' = xs1, z'' = xs2 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), splitmerge(1 + x'1 + (1 + x4 + xs3), 0, 0)) :|: x4 >= 0, z'' = 1 + x'1 + (1 + x4 + xs3), x'1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0, xs3 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 0) :|: z'' = 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 1 + x5 + 0) :|: x5 >= 0, xs'' >= 0, z'' = 1 + x5 + 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, splitmerge(1 + x'3 + (1 + x8 + xs5), 0, 0)) :|: x8 >= 0, x'3 >= 0, xs5 >= 0, z = 0, z'' = 1 + x'3 + (1 + x8 + xs5), z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 1 + x9 + 0) :|: z = 0, x9 >= 0, z' = 0, z'' = 1 + x9 + 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + x3 + 0, splitmerge(1 + x'2 + (1 + x6 + xs4), 0, 0)) :|: z'' = 1 + x'2 + (1 + x6 + xs4), x'2 >= 0, x6 >= 0, xs4 >= 0, z' = 1 + x3 + 0, z = 0, x3 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + x3 + 0, 0) :|: z'' = 0, z' = 1 + x3 + 0, z = 0, x3 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + x3 + 0, 1 + x7 + 0) :|: x7 >= 0, z'' = 1 + x7 + 0, z' = 1 + x3 + 0, z = 0, x3 >= 0 ---------------------------------------- (253) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (254) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> mergesort(z) :|: z >= 0 lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 1 }-> merge[Ite](lteq(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0 merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0 merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0 merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z'' >= 0, z' = 1 + x + xs, x >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, x >= 0, z' >= 0, z'' = 1 + x + xs mergesort(z) -{ 1 }-> splitmerge(1 + x' + (1 + x + xs), 0, 0) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) mergesort(z) -{ 1 }-> 0 :|: z = 0 mergesort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 splitmerge(z, z', z'') -{ 1 }-> splitmerge(xs, 1 + x + z'', z') :|: z = 1 + x + xs, xs >= 0, z'' >= 0, x >= 0, z' >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), splitmerge(1 + x'1 + (1 + x4 + xs3), 0, 0)) :|: x4 >= 0, z'' = 1 + x'1 + (1 + x4 + xs3), x'1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0, xs3 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 0) :|: z'' = 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, splitmerge(1 + x'3 + (1 + x8 + xs5), 0, 0)) :|: x8 >= 0, x'3 >= 0, xs5 >= 0, z = 0, z'' = 1 + x'3 + (1 + x8 + xs5), z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + x'2 + (1 + x6 + xs4), 0, 0)) :|: z'' = 1 + x'2 + (1 + x6 + xs4), x'2 >= 0, x6 >= 0, xs4 >= 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' - 1 >= 0 ---------------------------------------- (255) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { notEmpty } { lteq } { merge[Ite], merge } { splitmerge } { mergesort } { goal } ---------------------------------------- (256) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> mergesort(z) :|: z >= 0 lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 1 }-> merge[Ite](lteq(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0 merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0 merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0 merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z'' >= 0, z' = 1 + x + xs, x >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, x >= 0, z' >= 0, z'' = 1 + x + xs mergesort(z) -{ 1 }-> splitmerge(1 + x' + (1 + x + xs), 0, 0) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) mergesort(z) -{ 1 }-> 0 :|: z = 0 mergesort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 splitmerge(z, z', z'') -{ 1 }-> splitmerge(xs, 1 + x + z'', z') :|: z = 1 + x + xs, xs >= 0, z'' >= 0, x >= 0, z' >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), splitmerge(1 + x'1 + (1 + x4 + xs3), 0, 0)) :|: x4 >= 0, z'' = 1 + x'1 + (1 + x4 + xs3), x'1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0, xs3 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 0) :|: z'' = 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, splitmerge(1 + x'3 + (1 + x8 + xs5), 0, 0)) :|: x8 >= 0, x'3 >= 0, xs5 >= 0, z = 0, z'' = 1 + x'3 + (1 + x8 + xs5), z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + x'2 + (1 + x6 + xs4), 0, 0)) :|: z'' = 1 + x'2 + (1 + x6 + xs4), x'2 >= 0, x6 >= 0, xs4 >= 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' - 1 >= 0 Function symbols to be analyzed: {notEmpty}, {lteq}, {merge[Ite],merge}, {splitmerge}, {mergesort}, {goal} ---------------------------------------- (257) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (258) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> mergesort(z) :|: z >= 0 lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 1 }-> merge[Ite](lteq(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0 merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0 merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0 merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z'' >= 0, z' = 1 + x + xs, x >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, x >= 0, z' >= 0, z'' = 1 + x + xs mergesort(z) -{ 1 }-> splitmerge(1 + x' + (1 + x + xs), 0, 0) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) mergesort(z) -{ 1 }-> 0 :|: z = 0 mergesort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 splitmerge(z, z', z'') -{ 1 }-> splitmerge(xs, 1 + x + z'', z') :|: z = 1 + x + xs, xs >= 0, z'' >= 0, x >= 0, z' >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), splitmerge(1 + x'1 + (1 + x4 + xs3), 0, 0)) :|: x4 >= 0, z'' = 1 + x'1 + (1 + x4 + xs3), x'1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0, xs3 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 0) :|: z'' = 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, splitmerge(1 + x'3 + (1 + x8 + xs5), 0, 0)) :|: x8 >= 0, x'3 >= 0, xs5 >= 0, z = 0, z'' = 1 + x'3 + (1 + x8 + xs5), z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + x'2 + (1 + x6 + xs4), 0, 0)) :|: z'' = 1 + x'2 + (1 + x6 + xs4), x'2 >= 0, x6 >= 0, xs4 >= 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' - 1 >= 0 Function symbols to be analyzed: {notEmpty}, {lteq}, {merge[Ite],merge}, {splitmerge}, {mergesort}, {goal} ---------------------------------------- (259) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: notEmpty after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (260) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> mergesort(z) :|: z >= 0 lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 1 }-> merge[Ite](lteq(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0 merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0 merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0 merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z'' >= 0, z' = 1 + x + xs, x >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, x >= 0, z' >= 0, z'' = 1 + x + xs mergesort(z) -{ 1 }-> splitmerge(1 + x' + (1 + x + xs), 0, 0) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) mergesort(z) -{ 1 }-> 0 :|: z = 0 mergesort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 splitmerge(z, z', z'') -{ 1 }-> splitmerge(xs, 1 + x + z'', z') :|: z = 1 + x + xs, xs >= 0, z'' >= 0, x >= 0, z' >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), splitmerge(1 + x'1 + (1 + x4 + xs3), 0, 0)) :|: x4 >= 0, z'' = 1 + x'1 + (1 + x4 + xs3), x'1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0, xs3 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 0) :|: z'' = 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, splitmerge(1 + x'3 + (1 + x8 + xs5), 0, 0)) :|: x8 >= 0, x'3 >= 0, xs5 >= 0, z = 0, z'' = 1 + x'3 + (1 + x8 + xs5), z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + x'2 + (1 + x6 + xs4), 0, 0)) :|: z'' = 1 + x'2 + (1 + x6 + xs4), x'2 >= 0, x6 >= 0, xs4 >= 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' - 1 >= 0 Function symbols to be analyzed: {notEmpty}, {lteq}, {merge[Ite],merge}, {splitmerge}, {mergesort}, {goal} Previous analysis results are: notEmpty: runtime: ?, size: O(1) [2] ---------------------------------------- (261) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: notEmpty after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (262) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> mergesort(z) :|: z >= 0 lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 1 }-> merge[Ite](lteq(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0 merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0 merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0 merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z'' >= 0, z' = 1 + x + xs, x >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, x >= 0, z' >= 0, z'' = 1 + x + xs mergesort(z) -{ 1 }-> splitmerge(1 + x' + (1 + x + xs), 0, 0) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) mergesort(z) -{ 1 }-> 0 :|: z = 0 mergesort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 splitmerge(z, z', z'') -{ 1 }-> splitmerge(xs, 1 + x + z'', z') :|: z = 1 + x + xs, xs >= 0, z'' >= 0, x >= 0, z' >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), splitmerge(1 + x'1 + (1 + x4 + xs3), 0, 0)) :|: x4 >= 0, z'' = 1 + x'1 + (1 + x4 + xs3), x'1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0, xs3 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 0) :|: z'' = 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, splitmerge(1 + x'3 + (1 + x8 + xs5), 0, 0)) :|: x8 >= 0, x'3 >= 0, xs5 >= 0, z = 0, z'' = 1 + x'3 + (1 + x8 + xs5), z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + x'2 + (1 + x6 + xs4), 0, 0)) :|: z'' = 1 + x'2 + (1 + x6 + xs4), x'2 >= 0, x6 >= 0, xs4 >= 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' - 1 >= 0 Function symbols to be analyzed: {lteq}, {merge[Ite],merge}, {splitmerge}, {mergesort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (263) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (264) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> mergesort(z) :|: z >= 0 lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 1 }-> merge[Ite](lteq(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0 merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0 merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0 merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z'' >= 0, z' = 1 + x + xs, x >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, x >= 0, z' >= 0, z'' = 1 + x + xs mergesort(z) -{ 1 }-> splitmerge(1 + x' + (1 + x + xs), 0, 0) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) mergesort(z) -{ 1 }-> 0 :|: z = 0 mergesort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 splitmerge(z, z', z'') -{ 1 }-> splitmerge(xs, 1 + x + z'', z') :|: z = 1 + x + xs, xs >= 0, z'' >= 0, x >= 0, z' >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), splitmerge(1 + x'1 + (1 + x4 + xs3), 0, 0)) :|: x4 >= 0, z'' = 1 + x'1 + (1 + x4 + xs3), x'1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0, xs3 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 0) :|: z'' = 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, splitmerge(1 + x'3 + (1 + x8 + xs5), 0, 0)) :|: x8 >= 0, x'3 >= 0, xs5 >= 0, z = 0, z'' = 1 + x'3 + (1 + x8 + xs5), z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + x'2 + (1 + x6 + xs4), 0, 0)) :|: z'' = 1 + x'2 + (1 + x6 + xs4), x'2 >= 0, x6 >= 0, xs4 >= 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' - 1 >= 0 Function symbols to be analyzed: {lteq}, {merge[Ite],merge}, {splitmerge}, {mergesort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (265) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: lteq after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (266) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> mergesort(z) :|: z >= 0 lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 1 }-> merge[Ite](lteq(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0 merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0 merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0 merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z'' >= 0, z' = 1 + x + xs, x >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, x >= 0, z' >= 0, z'' = 1 + x + xs mergesort(z) -{ 1 }-> splitmerge(1 + x' + (1 + x + xs), 0, 0) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) mergesort(z) -{ 1 }-> 0 :|: z = 0 mergesort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 splitmerge(z, z', z'') -{ 1 }-> splitmerge(xs, 1 + x + z'', z') :|: z = 1 + x + xs, xs >= 0, z'' >= 0, x >= 0, z' >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), splitmerge(1 + x'1 + (1 + x4 + xs3), 0, 0)) :|: x4 >= 0, z'' = 1 + x'1 + (1 + x4 + xs3), x'1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0, xs3 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 0) :|: z'' = 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, splitmerge(1 + x'3 + (1 + x8 + xs5), 0, 0)) :|: x8 >= 0, x'3 >= 0, xs5 >= 0, z = 0, z'' = 1 + x'3 + (1 + x8 + xs5), z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + x'2 + (1 + x6 + xs4), 0, 0)) :|: z'' = 1 + x'2 + (1 + x6 + xs4), x'2 >= 0, x6 >= 0, xs4 >= 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' - 1 >= 0 Function symbols to be analyzed: {lteq}, {merge[Ite],merge}, {splitmerge}, {mergesort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] lteq: runtime: ?, size: O(1) [2] ---------------------------------------- (267) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: lteq after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (268) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> mergesort(z) :|: z >= 0 lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 1 }-> merge[Ite](lteq(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0 merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0 merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0 merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z'' >= 0, z' = 1 + x + xs, x >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, x >= 0, z' >= 0, z'' = 1 + x + xs mergesort(z) -{ 1 }-> splitmerge(1 + x' + (1 + x + xs), 0, 0) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) mergesort(z) -{ 1 }-> 0 :|: z = 0 mergesort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 splitmerge(z, z', z'') -{ 1 }-> splitmerge(xs, 1 + x + z'', z') :|: z = 1 + x + xs, xs >= 0, z'' >= 0, x >= 0, z' >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), splitmerge(1 + x'1 + (1 + x4 + xs3), 0, 0)) :|: x4 >= 0, z'' = 1 + x'1 + (1 + x4 + xs3), x'1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0, xs3 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 0) :|: z'' = 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, splitmerge(1 + x'3 + (1 + x8 + xs5), 0, 0)) :|: x8 >= 0, x'3 >= 0, xs5 >= 0, z = 0, z'' = 1 + x'3 + (1 + x8 + xs5), z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + x'2 + (1 + x6 + xs4), 0, 0)) :|: z'' = 1 + x'2 + (1 + x6 + xs4), x'2 >= 0, x6 >= 0, xs4 >= 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' - 1 >= 0 Function symbols to be analyzed: {merge[Ite],merge}, {splitmerge}, {mergesort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] lteq: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (269) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (270) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> mergesort(z) :|: z >= 0 lteq(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 1 }-> merge[Ite](s, 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: s >= 0, s <= 2, z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0 merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0 merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0 merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z'' >= 0, z' = 1 + x + xs, x >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, x >= 0, z' >= 0, z'' = 1 + x + xs mergesort(z) -{ 1 }-> splitmerge(1 + x' + (1 + x + xs), 0, 0) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) mergesort(z) -{ 1 }-> 0 :|: z = 0 mergesort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 splitmerge(z, z', z'') -{ 1 }-> splitmerge(xs, 1 + x + z'', z') :|: z = 1 + x + xs, xs >= 0, z'' >= 0, x >= 0, z' >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), splitmerge(1 + x'1 + (1 + x4 + xs3), 0, 0)) :|: x4 >= 0, z'' = 1 + x'1 + (1 + x4 + xs3), x'1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0, xs3 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 0) :|: z'' = 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, splitmerge(1 + x'3 + (1 + x8 + xs5), 0, 0)) :|: x8 >= 0, x'3 >= 0, xs5 >= 0, z = 0, z'' = 1 + x'3 + (1 + x8 + xs5), z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + x'2 + (1 + x6 + xs4), 0, 0)) :|: z'' = 1 + x'2 + (1 + x6 + xs4), x'2 >= 0, x6 >= 0, xs4 >= 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' - 1 >= 0 Function symbols to be analyzed: {merge[Ite],merge}, {splitmerge}, {mergesort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] lteq: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (271) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: merge[Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' + z'' Computed SIZE bound using CoFloCo for: merge after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (272) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> mergesort(z) :|: z >= 0 lteq(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 1 }-> merge[Ite](s, 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: s >= 0, s <= 2, z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0 merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0 merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0 merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z'' >= 0, z' = 1 + x + xs, x >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, x >= 0, z' >= 0, z'' = 1 + x + xs mergesort(z) -{ 1 }-> splitmerge(1 + x' + (1 + x + xs), 0, 0) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) mergesort(z) -{ 1 }-> 0 :|: z = 0 mergesort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 splitmerge(z, z', z'') -{ 1 }-> splitmerge(xs, 1 + x + z'', z') :|: z = 1 + x + xs, xs >= 0, z'' >= 0, x >= 0, z' >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), splitmerge(1 + x'1 + (1 + x4 + xs3), 0, 0)) :|: x4 >= 0, z'' = 1 + x'1 + (1 + x4 + xs3), x'1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0, xs3 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 0) :|: z'' = 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, splitmerge(1 + x'3 + (1 + x8 + xs5), 0, 0)) :|: x8 >= 0, x'3 >= 0, xs5 >= 0, z = 0, z'' = 1 + x'3 + (1 + x8 + xs5), z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + x'2 + (1 + x6 + xs4), 0, 0)) :|: z'' = 1 + x'2 + (1 + x6 + xs4), x'2 >= 0, x6 >= 0, xs4 >= 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' - 1 >= 0 Function symbols to be analyzed: {merge[Ite],merge}, {splitmerge}, {mergesort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] lteq: runtime: O(1) [0], size: O(1) [2] merge[Ite]: runtime: ?, size: O(n^1) [z' + z''] merge: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (273) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: merge[Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 8 + z' + z'' Computed RUNTIME bound using CoFloCo for: merge after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 9 + z + z' ---------------------------------------- (274) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> mergesort(z) :|: z >= 0 lteq(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 1 }-> merge[Ite](s, 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: s >= 0, s <= 2, z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0 merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0 merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0 merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z'' >= 0, z' = 1 + x + xs, x >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, x >= 0, z' >= 0, z'' = 1 + x + xs mergesort(z) -{ 1 }-> splitmerge(1 + x' + (1 + x + xs), 0, 0) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) mergesort(z) -{ 1 }-> 0 :|: z = 0 mergesort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 splitmerge(z, z', z'') -{ 1 }-> splitmerge(xs, 1 + x + z'', z') :|: z = 1 + x + xs, xs >= 0, z'' >= 0, x >= 0, z' >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), splitmerge(1 + x'1 + (1 + x4 + xs3), 0, 0)) :|: x4 >= 0, z'' = 1 + x'1 + (1 + x4 + xs3), x'1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0, xs3 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 0) :|: z'' = 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, splitmerge(1 + x'3 + (1 + x8 + xs5), 0, 0)) :|: x8 >= 0, x'3 >= 0, xs5 >= 0, z = 0, z'' = 1 + x'3 + (1 + x8 + xs5), z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 0) :|: z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, 1 + (z'' - 1) + 0) :|: z = 0, z'' - 1 >= 0, z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + x'2 + (1 + x6 + xs4), 0, 0)) :|: z'' = 1 + x'2 + (1 + x6 + xs4), x'2 >= 0, x6 >= 0, xs4 >= 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 0) :|: z'' = 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, z = 0, z' - 1 >= 0 Function symbols to be analyzed: {splitmerge}, {mergesort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] lteq: runtime: O(1) [0], size: O(1) [2] merge[Ite]: runtime: O(n^1) [8 + z' + z''], size: O(n^1) [z' + z''] merge: runtime: O(n^1) [9 + z + z'], size: O(n^1) [z + z'] ---------------------------------------- (275) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (276) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> mergesort(z) :|: z >= 0 lteq(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 13 + x'' + xs + xs' + y' }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + x'') + xs' + (1 + (1 + y') + xs), s >= 0, s <= 2, z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0 merge(z, z') -{ 10 + x + xs + z }-> s1 :|: s1 >= 0, s1 <= 1 + 0 + (z - 1) + (1 + x + xs), xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0 merge(z, z') -{ 11 + x1 + xs' + z' }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + x1) + xs' + (1 + 0 + (z' - 1)), z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0 merge(z, z') -{ 11 + x + x' + xs + xs' }-> s3 :|: s3 >= 0, s3 <= 1 + x' + xs' + (1 + x + xs), xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 9 + xs + z' }-> 1 + x + s8 :|: s8 >= 0, s8 <= z' + xs, xs >= 0, z = 1, x >= 0, z' >= 0, z'' = 1 + x + xs merge[Ite](z, z', z'') -{ 9 + xs + z'' }-> 1 + x + s9 :|: s9 >= 0, s9 <= xs + z'', z = 2, xs >= 0, z'' >= 0, z' = 1 + x + xs, x >= 0 mergesort(z) -{ 1 }-> splitmerge(1 + x' + (1 + x + xs), 0, 0) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) mergesort(z) -{ 1 }-> 0 :|: z = 0 mergesort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 splitmerge(z, z', z'') -{ 12 + z' + z'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z' - 1) + 0 + (1 + (z'' - 1) + 0), z'' - 1 >= 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 12 + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z' - 1) + 0 + 0, z'' = 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 12 + z'' }-> s6 :|: s6 >= 0, s6 <= 0 + (1 + (z'' - 1) + 0), z = 0, z'' - 1 >= 0, z' = 0 splitmerge(z, z', z'') -{ 12 }-> s7 :|: s7 >= 0, s7 <= 0 + 0, z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 1 }-> splitmerge(xs, 1 + x + z'', z') :|: z = 1 + x + xs, xs >= 0, z'' >= 0, x >= 0, z' >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), splitmerge(1 + x'1 + (1 + x4 + xs3), 0, 0)) :|: x4 >= 0, z'' = 1 + x'1 + (1 + x4 + xs3), x'1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0, xs3 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 0) :|: z'' = 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, splitmerge(1 + x'3 + (1 + x8 + xs5), 0, 0)) :|: x8 >= 0, x'3 >= 0, xs5 >= 0, z = 0, z'' = 1 + x'3 + (1 + x8 + xs5), z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + x'2 + (1 + x6 + xs4), 0, 0)) :|: z'' = 1 + x'2 + (1 + x6 + xs4), x'2 >= 0, x6 >= 0, xs4 >= 0, z = 0, z' - 1 >= 0 Function symbols to be analyzed: {splitmerge}, {mergesort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] lteq: runtime: O(1) [0], size: O(1) [2] merge[Ite]: runtime: O(n^1) [8 + z' + z''], size: O(n^1) [z' + z''] merge: runtime: O(n^1) [9 + z + z'], size: O(n^1) [z + z'] ---------------------------------------- (277) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: splitmerge after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 11 + 23*z + 23*z' + 23*z'' ---------------------------------------- (278) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> mergesort(z) :|: z >= 0 lteq(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 13 + x'' + xs + xs' + y' }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + x'') + xs' + (1 + (1 + y') + xs), s >= 0, s <= 2, z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0 merge(z, z') -{ 10 + x + xs + z }-> s1 :|: s1 >= 0, s1 <= 1 + 0 + (z - 1) + (1 + x + xs), xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0 merge(z, z') -{ 11 + x1 + xs' + z' }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + x1) + xs' + (1 + 0 + (z' - 1)), z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0 merge(z, z') -{ 11 + x + x' + xs + xs' }-> s3 :|: s3 >= 0, s3 <= 1 + x' + xs' + (1 + x + xs), xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 9 + xs + z' }-> 1 + x + s8 :|: s8 >= 0, s8 <= z' + xs, xs >= 0, z = 1, x >= 0, z' >= 0, z'' = 1 + x + xs merge[Ite](z, z', z'') -{ 9 + xs + z'' }-> 1 + x + s9 :|: s9 >= 0, s9 <= xs + z'', z = 2, xs >= 0, z'' >= 0, z' = 1 + x + xs, x >= 0 mergesort(z) -{ 1 }-> splitmerge(1 + x' + (1 + x + xs), 0, 0) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) mergesort(z) -{ 1 }-> 0 :|: z = 0 mergesort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 splitmerge(z, z', z'') -{ 12 + z' + z'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z' - 1) + 0 + (1 + (z'' - 1) + 0), z'' - 1 >= 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 12 + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z' - 1) + 0 + 0, z'' = 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 12 + z'' }-> s6 :|: s6 >= 0, s6 <= 0 + (1 + (z'' - 1) + 0), z = 0, z'' - 1 >= 0, z' = 0 splitmerge(z, z', z'') -{ 12 }-> s7 :|: s7 >= 0, s7 <= 0 + 0, z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 1 }-> splitmerge(xs, 1 + x + z'', z') :|: z = 1 + x + xs, xs >= 0, z'' >= 0, x >= 0, z' >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), splitmerge(1 + x'1 + (1 + x4 + xs3), 0, 0)) :|: x4 >= 0, z'' = 1 + x'1 + (1 + x4 + xs3), x'1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0, xs3 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 0) :|: z'' = 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, splitmerge(1 + x'3 + (1 + x8 + xs5), 0, 0)) :|: x8 >= 0, x'3 >= 0, xs5 >= 0, z = 0, z'' = 1 + x'3 + (1 + x8 + xs5), z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + x'2 + (1 + x6 + xs4), 0, 0)) :|: z'' = 1 + x'2 + (1 + x6 + xs4), x'2 >= 0, x6 >= 0, xs4 >= 0, z = 0, z' - 1 >= 0 Function symbols to be analyzed: {splitmerge}, {mergesort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] lteq: runtime: O(1) [0], size: O(1) [2] merge[Ite]: runtime: O(n^1) [8 + z' + z''], size: O(n^1) [z' + z''] merge: runtime: O(n^1) [9 + z + z'], size: O(n^1) [z + z'] splitmerge: runtime: ?, size: O(n^1) [11 + 23*z + 23*z' + 23*z''] ---------------------------------------- (279) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: splitmerge after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (280) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> mergesort(z) :|: z >= 0 lteq(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 merge(z, z') -{ 13 + x'' + xs + xs' + y' }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + x'') + xs' + (1 + (1 + y') + xs), s >= 0, s <= 2, z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0 merge(z, z') -{ 10 + x + xs + z }-> s1 :|: s1 >= 0, s1 <= 1 + 0 + (z - 1) + (1 + x + xs), xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0 merge(z, z') -{ 11 + x1 + xs' + z' }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + x1) + xs' + (1 + 0 + (z' - 1)), z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0 merge(z, z') -{ 11 + x + x' + xs + xs' }-> s3 :|: s3 >= 0, s3 <= 1 + x' + xs' + (1 + x + xs), xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 merge[Ite](z, z', z'') -{ 9 + xs + z' }-> 1 + x + s8 :|: s8 >= 0, s8 <= z' + xs, xs >= 0, z = 1, x >= 0, z' >= 0, z'' = 1 + x + xs merge[Ite](z, z', z'') -{ 9 + xs + z'' }-> 1 + x + s9 :|: s9 >= 0, s9 <= xs + z'', z = 2, xs >= 0, z'' >= 0, z' = 1 + x + xs, x >= 0 mergesort(z) -{ 1 }-> splitmerge(1 + x' + (1 + x + xs), 0, 0) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) mergesort(z) -{ 1 }-> 0 :|: z = 0 mergesort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 splitmerge(z, z', z'') -{ 12 + z' + z'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z' - 1) + 0 + (1 + (z'' - 1) + 0), z'' - 1 >= 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 12 + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z' - 1) + 0 + 0, z'' = 0, z = 0, z' - 1 >= 0 splitmerge(z, z', z'') -{ 12 + z'' }-> s6 :|: s6 >= 0, s6 <= 0 + (1 + (z'' - 1) + 0), z = 0, z'' - 1 >= 0, z' = 0 splitmerge(z, z', z'') -{ 12 }-> s7 :|: s7 >= 0, s7 <= 0 + 0, z'' = 0, z = 0, z' = 0 splitmerge(z, z', z'') -{ 1 }-> splitmerge(xs, 1 + x + z'', z') :|: z = 1 + x + xs, xs >= 0, z'' >= 0, x >= 0, z' >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), splitmerge(1 + x'1 + (1 + x4 + xs3), 0, 0)) :|: x4 >= 0, z'' = 1 + x'1 + (1 + x4 + xs3), x'1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0, xs3 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 0) :|: z'' = 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(splitmerge(1 + x''' + (1 + x2 + xs''), 0, 0), 1 + (z'' - 1) + 0) :|: z'' - 1 >= 0, xs'' >= 0, x''' >= 0, z = 0, z' = 1 + x''' + (1 + x2 + xs''), x2 >= 0 splitmerge(z, z', z'') -{ 3 }-> merge(0, splitmerge(1 + x'3 + (1 + x8 + xs5), 0, 0)) :|: x8 >= 0, x'3 >= 0, xs5 >= 0, z = 0, z'' = 1 + x'3 + (1 + x8 + xs5), z' = 0 splitmerge(z, z', z'') -{ 3 }-> merge(1 + (z' - 1) + 0, splitmerge(1 + x'2 + (1 + x6 + xs4), 0, 0)) :|: z'' = 1 + x'2 + (1 + x6 + xs4), x'2 >= 0, x6 >= 0, xs4 >= 0, z = 0, z' - 1 >= 0 Function symbols to be analyzed: {splitmerge}, {mergesort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] lteq: runtime: O(1) [0], size: O(1) [2] merge[Ite]: runtime: O(n^1) [8 + z' + z''], size: O(n^1) [z' + z''] merge: runtime: O(n^1) [9 + z + z'], size: O(n^1) [z + z'] splitmerge: runtime: INF, size: O(n^1) [11 + 23*z + 23*z' + 23*z''] ---------------------------------------- (281) 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: lteq(v0, v1) -> null_lteq [0] merge[Ite](v0, v1, v2) -> null_merge[Ite] [0] mergesort(v0) -> null_mergesort [0] merge(v0, v1) -> null_merge [0] splitmerge(v0, v1, v2) -> null_splitmerge [0] notEmpty(v0) -> null_notEmpty [0] And the following fresh constants: null_lteq, null_merge[Ite], null_mergesort, null_merge, null_splitmerge, null_notEmpty ---------------------------------------- (282) 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: mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil) [1] mergesort(Cons(x, Nil)) -> Cons(x, Nil) [1] merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](lteq(x', x), Cons(x', xs'), Cons(x, xs)) [1] merge(Cons(x, xs), Nil) -> Cons(x, xs) [1] splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1) [1] splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2)) [1] mergesort(Nil) -> Nil [1] merge(Nil, xs2) -> xs2 [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] goal(xs) -> mergesort(xs) [1] lteq(S(x), S(y)) -> lteq(x, y) [0] lteq(0, y) -> True [0] lteq(S(x), 0) -> False [0] merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs)) [0] merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2)) [0] lteq(v0, v1) -> null_lteq [0] merge[Ite](v0, v1, v2) -> null_merge[Ite] [0] mergesort(v0) -> null_mergesort [0] merge(v0, v1) -> null_merge [0] splitmerge(v0, v1, v2) -> null_splitmerge [0] notEmpty(v0) -> null_notEmpty [0] The TRS has the following type information: mergesort :: Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge -> Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge Cons :: S:0 -> Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge -> Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge splitmerge :: Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge -> Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge -> Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge -> Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge Nil :: Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge merge :: Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge -> Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge -> Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge merge[Ite] :: True:False:null_lteq:null_notEmpty -> Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge -> Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge -> Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge lteq :: S:0 -> S:0 -> True:False:null_lteq:null_notEmpty notEmpty :: Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge -> True:False:null_lteq:null_notEmpty True :: True:False:null_lteq:null_notEmpty False :: True:False:null_lteq:null_notEmpty goal :: Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge -> Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge S :: S:0 -> S:0 0 :: S:0 null_lteq :: True:False:null_lteq:null_notEmpty null_merge[Ite] :: Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge null_mergesort :: Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge null_merge :: Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge null_splitmerge :: Cons:Nil:null_merge[Ite]:null_mergesort:null_merge:null_splitmerge null_notEmpty :: True:False:null_lteq:null_notEmpty Rewrite Strategy: INNERMOST ---------------------------------------- (283) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: Nil => 0 True => 2 False => 1 0 => 0 null_lteq => 0 null_merge[Ite] => 0 null_mergesort => 0 null_merge => 0 null_splitmerge => 0 null_notEmpty => 0 ---------------------------------------- (284) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> mergesort(xs) :|: xs >= 0, z = xs lteq(z, z') -{ 0 }-> lteq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x lteq(z, z') -{ 0 }-> 2 :|: y >= 0, z = 0, z' = y lteq(z, z') -{ 0 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 lteq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 merge(z, z') -{ 1 }-> xs2 :|: xs2 >= 0, z' = xs2, z = 0 merge(z, z') -{ 1 }-> merge[Ite](lteq(x', x), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' merge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 merge[Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, xs2) :|: z = 2, xs >= 0, xs2 >= 0, z' = 1 + x + xs, x >= 0, z'' = xs2 merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs1, xs) :|: xs >= 0, z = 1, x >= 0, xs1 >= 0, z'' = 1 + x + xs, z' = xs1 mergesort(z) -{ 1 }-> splitmerge(1 + x' + (1 + x + xs), 0, 0) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) mergesort(z) -{ 1 }-> 0 :|: z = 0 mergesort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 mergesort(z) -{ 1 }-> 1 + x + 0 :|: x >= 0, z = 1 + x + 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 notEmpty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 splitmerge(z, z', z'') -{ 1 }-> splitmerge(xs, 1 + x + xs2, xs1) :|: z = 1 + x + xs, xs >= 0, xs2 >= 0, x >= 0, xs1 >= 0, z' = xs1, z'' = xs2 splitmerge(z, z', z'') -{ 1 }-> merge(mergesort(xs1), mergesort(xs2)) :|: xs2 >= 0, xs1 >= 0, z = 0, z' = xs1, z'' = xs2 splitmerge(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 Only complete derivations are relevant for the runtime complexity.