KILLED proof of input_eVPq8ySgJC.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).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), 216 ms] (2) CpxRelTRS (3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (8) CpxWeightedTrs (9) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxWeightedTrs (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedTrs (13) CompletionProof [UPPER BOUND(ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxTypedWeightedCompleteTrs (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 188 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 9 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 348 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 107 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 316 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 76 ms] (40) CpxRNTS (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 323 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 65 ms] (46) CpxRNTS (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 914 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 267 ms] (52) CpxRNTS (53) ResultPropagationProof [UPPER BOUND(ID), 2 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 2184 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 1009 ms] (58) CpxRNTS (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 4233 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 568 ms] (64) CpxRNTS (65) CompletionProof [UPPER BOUND(ID), 0 ms] (66) CpxTypedWeightedCompleteTrs (67) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 3 ms] (68) CpxRNTS (69) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (70) CdtProblem (71) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (72) CdtProblem (73) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 218 ms] (76) CdtProblem (77) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 94 ms] (80) CdtProblem (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 84 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 81 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 93 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtForwardInstantiationProof [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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtRewritingProof [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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtRewritingProof [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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem ---------------------------------------- (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: quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) quicksort(Cons(x, Nil)) -> Cons(x, Nil) quicksort(Nil) -> Nil partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs)) partLt(x, Nil) -> Nil partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs)) partGt(x, Nil) -> Nil app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) app(Nil, ys) -> ys notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) goal(xs) -> quicksort(xs) The (relative) TRS S consists of the following rules: <(S(x), S(y)) -> <(x, y) <(0, S(y)) -> True <(x, 0) -> False >(S(x), S(y)) -> >(x, y) >(0, y) -> False >(S(x), 0) -> True partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) 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: quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) quicksort(Cons(x, Nil)) -> Cons(x, Nil) quicksort(Nil) -> Nil partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs)) partLt(x, Nil) -> Nil partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs)) partGt(x, Nil) -> Nil app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) app(Nil, ys) -> ys notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) goal(xs) -> quicksort(xs) The (relative) TRS S consists of the following rules: <(S(x), S(y)) -> <(x, y) <(0, S(y)) -> True <(x, 0) -> False >(S(x), S(y)) -> >(x, y) >(0, y) -> False >(S(x), 0) -> True partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) 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: quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) quicksort(Cons(x, Nil)) -> Cons(x, Nil) quicksort(Nil) -> Nil partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs)) partLt(x, Nil) -> Nil partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs)) partGt(x, Nil) -> Nil app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) app(Nil, ys) -> ys notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) goal(xs) -> quicksort(xs) The (relative) TRS S consists of the following rules: <(S(x), S(y)) -> <(x, y) <(0', S(y)) -> True <(x, 0') -> False >(S(x), S(y)) -> >(x, y) >(0', y) -> False >(S(x), 0') -> True partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) 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: quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) quicksort(Cons(x, Nil)) -> Cons(x, Nil) quicksort(Nil) -> Nil partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs)) partLt(x, Nil) -> Nil partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs)) partGt(x, Nil) -> Nil app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) app(Nil, ys) -> ys notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) goal(xs) -> quicksort(xs) <(S(x), S(y)) -> <(x, y) <(0, S(y)) -> True <(x, 0) -> False >(S(x), S(y)) -> >(x, y) >(0, y) -> False >(S(x), 0) -> True partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (7) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (8) 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: quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) [1] quicksort(Cons(x, Nil)) -> Cons(x, Nil) [1] quicksort(Nil) -> Nil [1] partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs)) [1] partLt(x, Nil) -> Nil [1] partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs)) [1] partGt(x, Nil) -> Nil [1] app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) [1] app(Nil, ys) -> ys [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) [1] goal(xs) -> quicksort(xs) [1] <(S(x), S(y)) -> <(x, y) [0] <(0, S(y)) -> True [0] <(x, 0) -> False [0] >(S(x), S(y)) -> >(x, y) [0] >(0, y) -> False [0] >(S(x), 0) -> True [0] partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) [0] partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) [0] partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) [0] partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (9) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: < => lt > => gr ---------------------------------------- (10) 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: quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) [1] quicksort(Cons(x, Nil)) -> Cons(x, Nil) [1] quicksort(Nil) -> Nil [1] partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](lt(x, x'), x', Cons(x, xs)) [1] partLt(x, Nil) -> Nil [1] partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](gr(x, x'), x', Cons(x, xs)) [1] partGt(x, Nil) -> Nil [1] app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) [1] app(Nil, ys) -> ys [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) [1] goal(xs) -> quicksort(xs) [1] lt(S(x), S(y)) -> lt(x, y) [0] lt(0, S(y)) -> True [0] lt(x, 0) -> False [0] gr(S(x), S(y)) -> gr(x, y) [0] gr(0, y) -> False [0] gr(S(x), 0) -> True [0] partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) [0] partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) [0] partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) [0] partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (11) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) [1] quicksort(Cons(x, Nil)) -> Cons(x, Nil) [1] quicksort(Nil) -> Nil [1] partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](lt(x, x'), x', Cons(x, xs)) [1] partLt(x, Nil) -> Nil [1] partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](gr(x, x'), x', Cons(x, xs)) [1] partGt(x, Nil) -> Nil [1] app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) [1] app(Nil, ys) -> ys [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) [1] goal(xs) -> quicksort(xs) [1] lt(S(x), S(y)) -> lt(x, y) [0] lt(0, S(y)) -> True [0] lt(x, 0) -> False [0] gr(S(x), S(y)) -> gr(x, y) [0] gr(0, y) -> False [0] gr(S(x), 0) -> True [0] partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) [0] partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) [0] partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) [0] partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) [0] The TRS has the following type information: quicksort :: Cons:Nil -> Cons:Nil Cons :: S:0 -> Cons:Nil -> Cons:Nil part :: S:0 -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil partLt :: S:0 -> Cons:Nil -> Cons:Nil partLt[Ite][True][Ite] :: True:False -> S:0 -> Cons:Nil -> Cons:Nil lt :: S:0 -> S:0 -> True:False partGt :: S:0 -> Cons:Nil -> Cons:Nil partGt[Ite][True][Ite] :: True:False -> S:0 -> Cons:Nil -> Cons:Nil gr :: S:0 -> S:0 -> True:False app :: Cons:Nil -> Cons:Nil -> Cons:Nil 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 ---------------------------------------- (13) 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: quicksort_1 partLt_2 partGt_2 part_2 app_2 lt_2 gr_2 partLt[Ite][True][Ite]_3 partGt[Ite][True][Ite]_3 Due to the following rules being added: lt(v0, v1) -> null_lt [0] gr(v0, v1) -> null_gr [0] partLt[Ite][True][Ite](v0, v1, v2) -> Nil [0] partGt[Ite][True][Ite](v0, v1, v2) -> Nil [0] And the following fresh constants: null_lt, null_gr ---------------------------------------- (14) 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: quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) [1] quicksort(Cons(x, Nil)) -> Cons(x, Nil) [1] quicksort(Nil) -> Nil [1] partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](lt(x, x'), x', Cons(x, xs)) [1] partLt(x, Nil) -> Nil [1] partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](gr(x, x'), x', Cons(x, xs)) [1] partGt(x, Nil) -> Nil [1] app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) [1] app(Nil, ys) -> ys [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) [1] goal(xs) -> quicksort(xs) [1] lt(S(x), S(y)) -> lt(x, y) [0] lt(0, S(y)) -> True [0] lt(x, 0) -> False [0] gr(S(x), S(y)) -> gr(x, y) [0] gr(0, y) -> False [0] gr(S(x), 0) -> True [0] partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) [0] partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) [0] partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) [0] partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) [0] lt(v0, v1) -> null_lt [0] gr(v0, v1) -> null_gr [0] partLt[Ite][True][Ite](v0, v1, v2) -> Nil [0] partGt[Ite][True][Ite](v0, v1, v2) -> Nil [0] The TRS has the following type information: quicksort :: Cons:Nil -> Cons:Nil Cons :: S:0 -> Cons:Nil -> Cons:Nil part :: S:0 -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil partLt :: S:0 -> Cons:Nil -> Cons:Nil partLt[Ite][True][Ite] :: True:False:null_lt:null_gr -> S:0 -> Cons:Nil -> Cons:Nil lt :: S:0 -> S:0 -> True:False:null_lt:null_gr partGt :: S:0 -> Cons:Nil -> Cons:Nil partGt[Ite][True][Ite] :: True:False:null_lt:null_gr -> S:0 -> Cons:Nil -> Cons:Nil gr :: S:0 -> S:0 -> True:False:null_lt:null_gr app :: Cons:Nil -> Cons:Nil -> Cons:Nil notEmpty :: Cons:Nil -> True:False:null_lt:null_gr True :: True:False:null_lt:null_gr False :: True:False:null_lt:null_gr goal :: Cons:Nil -> Cons:Nil S :: S:0 -> S:0 0 :: S:0 null_lt :: True:False:null_lt:null_gr null_gr :: True:False:null_lt:null_gr Rewrite Strategy: INNERMOST ---------------------------------------- (15) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (16) 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: quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) [1] quicksort(Cons(x, Nil)) -> Cons(x, Nil) [1] quicksort(Nil) -> Nil [1] partLt(S(y'), Cons(S(x''), xs)) -> partLt[Ite][True][Ite](lt(x'', y'), S(y'), Cons(S(x''), xs)) [1] partLt(S(y''), Cons(0, xs)) -> partLt[Ite][True][Ite](True, S(y''), Cons(0, xs)) [1] partLt(0, Cons(x, xs)) -> partLt[Ite][True][Ite](False, 0, Cons(x, xs)) [1] partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](null_lt, x', Cons(x, xs)) [1] partLt(x, Nil) -> Nil [1] partGt(S(y1), Cons(S(x1), xs)) -> partGt[Ite][True][Ite](gr(x1, y1), S(y1), Cons(S(x1), xs)) [1] partGt(x', Cons(0, xs)) -> partGt[Ite][True][Ite](False, x', Cons(0, xs)) [1] partGt(0, Cons(S(x2), xs)) -> partGt[Ite][True][Ite](True, 0, Cons(S(x2), xs)) [1] partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](null_gr, x', Cons(x, xs)) [1] partGt(x, Nil) -> Nil [1] app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) [1] app(Nil, ys) -> ys [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] part(x, Cons(x3, xs')) -> app(quicksort(partLt[Ite][True][Ite](lt(x3, x), x, Cons(x3, xs'))), Cons(x, quicksort(partGt[Ite][True][Ite](gr(x3, x), x, Cons(x3, xs'))))) [3] part(x, Nil) -> app(quicksort(Nil), Cons(x, quicksort(Nil))) [3] goal(xs) -> quicksort(xs) [1] lt(S(x), S(y)) -> lt(x, y) [0] lt(0, S(y)) -> True [0] lt(x, 0) -> False [0] gr(S(x), S(y)) -> gr(x, y) [0] gr(0, y) -> False [0] gr(S(x), 0) -> True [0] partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) [0] partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) [0] partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) [0] partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) [0] lt(v0, v1) -> null_lt [0] gr(v0, v1) -> null_gr [0] partLt[Ite][True][Ite](v0, v1, v2) -> Nil [0] partGt[Ite][True][Ite](v0, v1, v2) -> Nil [0] The TRS has the following type information: quicksort :: Cons:Nil -> Cons:Nil Cons :: S:0 -> Cons:Nil -> Cons:Nil part :: S:0 -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil partLt :: S:0 -> Cons:Nil -> Cons:Nil partLt[Ite][True][Ite] :: True:False:null_lt:null_gr -> S:0 -> Cons:Nil -> Cons:Nil lt :: S:0 -> S:0 -> True:False:null_lt:null_gr partGt :: S:0 -> Cons:Nil -> Cons:Nil partGt[Ite][True][Ite] :: True:False:null_lt:null_gr -> S:0 -> Cons:Nil -> Cons:Nil gr :: S:0 -> S:0 -> True:False:null_lt:null_gr app :: Cons:Nil -> Cons:Nil -> Cons:Nil notEmpty :: Cons:Nil -> True:False:null_lt:null_gr True :: True:False:null_lt:null_gr False :: True:False:null_lt:null_gr goal :: Cons:Nil -> Cons:Nil S :: S:0 -> S:0 0 :: S:0 null_lt :: True:False:null_lt:null_gr null_gr :: True:False:null_lt:null_gr Rewrite Strategy: INNERMOST ---------------------------------------- (17) 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_lt => 0 null_gr => 0 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> ys :|: z' = ys, ys >= 0, z = 0 app(z, z') -{ 1 }-> 1 + x + app(xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(xs) :|: xs >= 0, z = xs gr(z, z') -{ 0 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x gr(z, z') -{ 0 }-> 2 :|: x >= 0, z = 1 + x, z' = 0 gr(z, z') -{ 0 }-> 1 :|: y >= 0, z = 0, z' = y gr(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 lt(z, z') -{ 0 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x lt(z, z') -{ 0 }-> 2 :|: z' = 1 + y, y >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: x >= 0, z = x, z' = 0 lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, x), x, 1 + x3 + xs')), 1 + x + quicksort(partGt[Ite][True][Ite](gr(x3, x), x, 1 + x3 + xs'))) :|: z' = 1 + x3 + xs', x >= 0, xs' >= 0, z = x, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + x + quicksort(0)) :|: x >= 0, z = x, z' = 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x1, y1), 1 + y1, 1 + (1 + x1) + xs) :|: z = 1 + y1, y1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, x', 1 + 0 + xs) :|: xs >= 0, z' = 1 + 0 + xs, x' >= 0, z = x' partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x' partGt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(x', xs) :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', y'), 1 + y', 1 + (1 + x'') + xs) :|: xs >= 0, z = 1 + y', y' >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + y'', 1 + 0 + xs) :|: xs >= 0, z = 1 + y'', z' = 1 + 0 + xs, y'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x' partLt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(x', xs) :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + x + 0 :|: x >= 0, z = 1 + x + 0 ---------------------------------------- (19) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](gr(x3, z), z, 1 + x3 + xs'))) :|: z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + xs) :|: z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 ---------------------------------------- (21) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { notEmpty } { gr } { lt } { app } { partGt, partGt[Ite][True][Ite] } { partLt[Ite][True][Ite], partLt } { part, quicksort } { goal } ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](gr(x3, z), z, 1 + x3 + xs'))) :|: z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + xs) :|: z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {notEmpty}, {gr}, {lt}, {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal} ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](gr(x3, z), z, 1 + x3 + xs'))) :|: z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + xs) :|: z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {notEmpty}, {gr}, {lt}, {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal} ---------------------------------------- (25) 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 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](gr(x3, z), z, 1 + x3 + xs'))) :|: z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + xs) :|: z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {notEmpty}, {gr}, {lt}, {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: ?, size: O(1) [2] ---------------------------------------- (27) 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 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](gr(x3, z), z, 1 + x3 + xs'))) :|: z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + xs) :|: z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {gr}, {lt}, {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (29) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](gr(x3, z), z, 1 + x3 + xs'))) :|: z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + xs) :|: z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {gr}, {lt}, {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: gr after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](gr(x3, z), z, 1 + x3 + xs'))) :|: z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + xs) :|: z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {gr}, {lt}, {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] gr: runtime: ?, size: O(1) [2] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: gr after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](gr(x3, z), z, 1 + x3 + xs'))) :|: z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + xs) :|: z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {lt}, {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] gr: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (35) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](s', z, 1 + x3 + xs'))) :|: s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + x1) + xs) :|: s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {lt}, {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] gr: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: lt after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](s', z, 1 + x3 + xs'))) :|: s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + x1) + xs) :|: s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {lt}, {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] gr: runtime: O(1) [0], size: O(1) [2] lt: runtime: ?, size: O(1) [2] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: lt after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](s', z, 1 + x3 + xs'))) :|: s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + x1) + xs) :|: s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] gr: runtime: O(1) [0], size: O(1) [2] lt: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (41) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](s2, z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](s', z, 1 + x3 + xs'))) :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + x1) + xs) :|: s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs) :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] gr: runtime: O(1) [0], size: O(1) [2] lt: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: app after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](s2, z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](s', z, 1 + x3 + xs'))) :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + x1) + xs) :|: s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs) :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] gr: runtime: O(1) [0], size: O(1) [2] lt: runtime: O(1) [0], size: O(1) [2] app: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: app after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](s2, z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](s', z, 1 + x3 + xs'))) :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + x1) + xs) :|: s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs) :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] gr: runtime: O(1) [0], size: O(1) [2] lt: runtime: O(1) [0], size: O(1) [2] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (47) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](s2, z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](s', z, 1 + x3 + xs'))) :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + x1) + xs) :|: s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs) :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] gr: runtime: O(1) [0], size: O(1) [2] lt: runtime: O(1) [0], size: O(1) [2] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: partGt after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' Computed SIZE bound using KoAT for: partGt[Ite][True][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z'' ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](s2, z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](s', z, 1 + x3 + xs'))) :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + x1) + xs) :|: s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs) :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] gr: runtime: O(1) [0], size: O(1) [2] lt: runtime: O(1) [0], size: O(1) [2] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] partGt: runtime: ?, size: O(n^1) [z'] partGt[Ite][True][Ite]: runtime: ?, size: O(n^1) [z''] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: partGt after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' Computed RUNTIME bound using CoFloCo for: partGt[Ite][True][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z'' ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](s2, z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](s', z, 1 + x3 + xs'))) :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + x1) + xs) :|: s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs) :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] gr: runtime: O(1) [0], size: O(1) [2] lt: runtime: O(1) [0], size: O(1) [2] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] partGt: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] partGt[Ite][True][Ite]: runtime: O(n^1) [z''], size: O(n^1) [z''] ---------------------------------------- (53) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 4 + x3 + xs' }-> app(quicksort(partLt[Ite][True][Ite](s2, z, 1 + x3 + xs')), 1 + z + quicksort(s9)) :|: s9 >= 0, s9 <= 1 + x3 + xs', s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 3 + x1 + xs }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + x1) + xs, s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 + z' }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0 partGt(z, z') -{ 3 + x2 + xs }-> s7 :|: s7 >= 0, s7 <= 1 + (1 + x2) + xs, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 2 + x + xs }-> s8 :|: s8 >= 0, s8 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> s11 :|: s11 >= 0, s11 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> 1 + x + s10 :|: s10 >= 0, s10 <= xs, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs) :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] gr: runtime: O(1) [0], size: O(1) [2] lt: runtime: O(1) [0], size: O(1) [2] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] partGt: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] partGt[Ite][True][Ite]: runtime: O(n^1) [z''], size: O(n^1) [z''] ---------------------------------------- (55) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: partLt[Ite][True][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z'' Computed SIZE bound using CoFloCo for: partLt after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 4 + x3 + xs' }-> app(quicksort(partLt[Ite][True][Ite](s2, z, 1 + x3 + xs')), 1 + z + quicksort(s9)) :|: s9 >= 0, s9 <= 1 + x3 + xs', s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 3 + x1 + xs }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + x1) + xs, s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 + z' }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0 partGt(z, z') -{ 3 + x2 + xs }-> s7 :|: s7 >= 0, s7 <= 1 + (1 + x2) + xs, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 2 + x + xs }-> s8 :|: s8 >= 0, s8 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> s11 :|: s11 >= 0, s11 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> 1 + x + s10 :|: s10 >= 0, s10 <= xs, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs) :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] gr: runtime: O(1) [0], size: O(1) [2] lt: runtime: O(1) [0], size: O(1) [2] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] partGt: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] partGt[Ite][True][Ite]: runtime: O(n^1) [z''], size: O(n^1) [z''] partLt[Ite][True][Ite]: runtime: ?, size: O(n^1) [z''] partLt: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: partLt[Ite][True][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 5 + z'' Computed RUNTIME bound using CoFloCo for: partLt after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 6 + z' ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 4 + x3 + xs' }-> app(quicksort(partLt[Ite][True][Ite](s2, z, 1 + x3 + xs')), 1 + z + quicksort(s9)) :|: s9 >= 0, s9 <= 1 + x3 + xs', s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 3 + x1 + xs }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + x1) + xs, s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 + z' }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0 partGt(z, z') -{ 3 + x2 + xs }-> s7 :|: s7 >= 0, s7 <= 1 + (1 + x2) + xs, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 2 + x + xs }-> s8 :|: s8 >= 0, s8 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> s11 :|: s11 >= 0, s11 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> 1 + x + s10 :|: s10 >= 0, s10 <= xs, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs) :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] gr: runtime: O(1) [0], size: O(1) [2] lt: runtime: O(1) [0], size: O(1) [2] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] partGt: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] partGt[Ite][True][Ite]: runtime: O(n^1) [z''], size: O(n^1) [z''] partLt[Ite][True][Ite]: runtime: O(n^1) [5 + z''], size: O(n^1) [z''] partLt: runtime: O(n^1) [6 + z'], size: O(n^1) [z'] ---------------------------------------- (59) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 10 + 2*x3 + 2*xs' }-> app(quicksort(s16), 1 + z + quicksort(s9)) :|: s16 >= 0, s16 <= 1 + x3 + xs', s9 >= 0, s9 <= 1 + x3 + xs', s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 3 + x1 + xs }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + x1) + xs, s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 + z' }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0 partGt(z, z') -{ 3 + x2 + xs }-> s7 :|: s7 >= 0, s7 <= 1 + (1 + x2) + xs, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 2 + x + xs }-> s8 :|: s8 >= 0, s8 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> s11 :|: s11 >= 0, s11 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> 1 + x + s10 :|: s10 >= 0, s10 <= xs, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 8 + x'' + xs }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + x'') + xs, s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 6 + z' }-> s13 :|: s13 >= 0, s13 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 7 + x + xs }-> s14 :|: s14 >= 0, s14 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 7 + x + xs }-> s15 :|: s15 >= 0, s15 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 6 + xs }-> s18 :|: s18 >= 0, s18 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 6 + xs }-> 1 + x + s17 :|: s17 >= 0, s17 <= xs, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] gr: runtime: O(1) [0], size: O(1) [2] lt: runtime: O(1) [0], size: O(1) [2] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] partGt: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] partGt[Ite][True][Ite]: runtime: O(n^1) [z''], size: O(n^1) [z''] partLt[Ite][True][Ite]: runtime: O(n^1) [5 + z''], size: O(n^1) [z''] partLt: runtime: O(n^1) [6 + z'], size: O(n^1) [z'] ---------------------------------------- (61) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: part after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? Computed SIZE bound using CoFloCo for: quicksort after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 10 + 2*x3 + 2*xs' }-> app(quicksort(s16), 1 + z + quicksort(s9)) :|: s16 >= 0, s16 <= 1 + x3 + xs', s9 >= 0, s9 <= 1 + x3 + xs', s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 3 + x1 + xs }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + x1) + xs, s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 + z' }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0 partGt(z, z') -{ 3 + x2 + xs }-> s7 :|: s7 >= 0, s7 <= 1 + (1 + x2) + xs, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 2 + x + xs }-> s8 :|: s8 >= 0, s8 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> s11 :|: s11 >= 0, s11 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> 1 + x + s10 :|: s10 >= 0, s10 <= xs, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 8 + x'' + xs }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + x'') + xs, s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 6 + z' }-> s13 :|: s13 >= 0, s13 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 7 + x + xs }-> s14 :|: s14 >= 0, s14 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 7 + x + xs }-> s15 :|: s15 >= 0, s15 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 6 + xs }-> s18 :|: s18 >= 0, s18 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 6 + xs }-> 1 + x + s17 :|: s17 >= 0, s17 <= xs, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] gr: runtime: O(1) [0], size: O(1) [2] lt: runtime: O(1) [0], size: O(1) [2] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] partGt: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] partGt[Ite][True][Ite]: runtime: O(n^1) [z''], size: O(n^1) [z''] partLt[Ite][True][Ite]: runtime: O(n^1) [5 + z''], size: O(n^1) [z''] partLt: runtime: O(n^1) [6 + z'], size: O(n^1) [z'] part: runtime: ?, size: EXP quicksort: runtime: ?, size: INF ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: part after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(z) :|: z >= 0 gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 part(z, z') -{ 10 + 2*x3 + 2*xs' }-> app(quicksort(s16), 1 + z + quicksort(s9)) :|: s16 >= 0, s16 <= 1 + x3 + xs', s9 >= 0, s9 <= 1 + x3 + xs', s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0 part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0 partGt(z, z') -{ 3 + x1 + xs }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + x1) + xs, s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs partGt(z, z') -{ 1 + z' }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0 partGt(z, z') -{ 3 + x2 + xs }-> s7 :|: s7 >= 0, s7 <= 1 + (1 + x2) + xs, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 partGt(z, z') -{ 2 + x + xs }-> s8 :|: s8 >= 0, s8 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> s11 :|: s11 >= 0, s11 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> 1 + x + s10 :|: s10 >= 0, s10 <= xs, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 8 + x'' + xs }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + x'') + xs, s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 partLt(z, z') -{ 6 + z' }-> s13 :|: s13 >= 0, s13 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 partLt(z, z') -{ 7 + x + xs }-> s14 :|: s14 >= 0, s14 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 partLt(z, z') -{ 7 + x + xs }-> s15 :|: s15 >= 0, s15 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 partLt[Ite][True][Ite](z, z', z'') -{ 6 + xs }-> s18 :|: s18 >= 0, s18 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 6 + xs }-> 1 + x + s17 :|: s17 >= 0, s17 <= xs, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0 Function symbols to be analyzed: {part,quicksort}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] gr: runtime: O(1) [0], size: O(1) [2] lt: runtime: O(1) [0], size: O(1) [2] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] partGt: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] partGt[Ite][True][Ite]: runtime: O(n^1) [z''], size: O(n^1) [z''] partLt[Ite][True][Ite]: runtime: O(n^1) [5 + z''], size: O(n^1) [z''] partLt: runtime: O(n^1) [6 + z'], size: O(n^1) [z'] part: runtime: INF, size: EXP quicksort: runtime: ?, size: INF ---------------------------------------- (65) 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: lt(v0, v1) -> null_lt [0] gr(v0, v1) -> null_gr [0] partLt[Ite][True][Ite](v0, v1, v2) -> null_partLt[Ite][True][Ite] [0] partGt[Ite][True][Ite](v0, v1, v2) -> null_partGt[Ite][True][Ite] [0] quicksort(v0) -> null_quicksort [0] partLt(v0, v1) -> null_partLt [0] partGt(v0, v1) -> null_partGt [0] app(v0, v1) -> null_app [0] notEmpty(v0) -> null_notEmpty [0] And the following fresh constants: null_lt, null_gr, null_partLt[Ite][True][Ite], null_partGt[Ite][True][Ite], null_quicksort, null_partLt, null_partGt, null_app, null_notEmpty ---------------------------------------- (66) 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: quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) [1] quicksort(Cons(x, Nil)) -> Cons(x, Nil) [1] quicksort(Nil) -> Nil [1] partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](lt(x, x'), x', Cons(x, xs)) [1] partLt(x, Nil) -> Nil [1] partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](gr(x, x'), x', Cons(x, xs)) [1] partGt(x, Nil) -> Nil [1] app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) [1] app(Nil, ys) -> ys [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) [1] goal(xs) -> quicksort(xs) [1] lt(S(x), S(y)) -> lt(x, y) [0] lt(0, S(y)) -> True [0] lt(x, 0) -> False [0] gr(S(x), S(y)) -> gr(x, y) [0] gr(0, y) -> False [0] gr(S(x), 0) -> True [0] partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) [0] partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) [0] partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) [0] partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) [0] lt(v0, v1) -> null_lt [0] gr(v0, v1) -> null_gr [0] partLt[Ite][True][Ite](v0, v1, v2) -> null_partLt[Ite][True][Ite] [0] partGt[Ite][True][Ite](v0, v1, v2) -> null_partGt[Ite][True][Ite] [0] quicksort(v0) -> null_quicksort [0] partLt(v0, v1) -> null_partLt [0] partGt(v0, v1) -> null_partGt [0] app(v0, v1) -> null_app [0] notEmpty(v0) -> null_notEmpty [0] The TRS has the following type information: quicksort :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app Cons :: S:0 -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app part :: S:0 -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app Nil :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app partLt :: S:0 -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app partLt[Ite][True][Ite] :: True:False:null_lt:null_gr:null_notEmpty -> S:0 -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app lt :: S:0 -> S:0 -> True:False:null_lt:null_gr:null_notEmpty partGt :: S:0 -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app partGt[Ite][True][Ite] :: True:False:null_lt:null_gr:null_notEmpty -> S:0 -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app gr :: S:0 -> S:0 -> True:False:null_lt:null_gr:null_notEmpty app :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app notEmpty :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> True:False:null_lt:null_gr:null_notEmpty True :: True:False:null_lt:null_gr:null_notEmpty False :: True:False:null_lt:null_gr:null_notEmpty goal :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app S :: S:0 -> S:0 0 :: S:0 null_lt :: True:False:null_lt:null_gr:null_notEmpty null_gr :: True:False:null_lt:null_gr:null_notEmpty null_partLt[Ite][True][Ite] :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app null_partGt[Ite][True][Ite] :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app null_quicksort :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app null_partLt :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app null_partGt :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app null_app :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app null_notEmpty :: True:False:null_lt:null_gr:null_notEmpty Rewrite Strategy: INNERMOST ---------------------------------------- (67) 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_lt => 0 null_gr => 0 null_partLt[Ite][True][Ite] => 0 null_partGt[Ite][True][Ite] => 0 null_quicksort => 0 null_partLt => 0 null_partGt => 0 null_app => 0 null_notEmpty => 0 ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> ys :|: z' = ys, ys >= 0, z = 0 app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 app(z, z') -{ 1 }-> 1 + x + app(xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0 goal(z) -{ 1 }-> quicksort(xs) :|: xs >= 0, z = xs gr(z, z') -{ 0 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x gr(z, z') -{ 0 }-> 2 :|: x >= 0, z = 1 + x, z' = 0 gr(z, z') -{ 0 }-> 1 :|: y >= 0, z = 0, z' = y gr(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 lt(z, z') -{ 0 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x lt(z, z') -{ 0 }-> 2 :|: z' = 1 + y, y >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: x >= 0, z = x, z' = 0 lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 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 part(z, z') -{ 1 }-> app(quicksort(partLt(x, xs)), 1 + x + quicksort(partGt(x, xs))) :|: xs >= 0, x >= 0, z' = xs, z = x partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x, x'), x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x' partGt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 partGt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(x', xs) :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x, x'), x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x' partLt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 partLt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(x', xs) :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs) quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 quicksort(z) -{ 1 }-> 1 + x + 0 :|: x >= 0, z = 1 + x + 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (69) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) goal(z0) -> quicksort(z0) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) <'(0, S(z0)) -> c1 <'(z0, 0) -> c2 >'(S(z0), S(z1)) -> c3(>'(z0, z1)) >'(0, z0) -> c4 >'(S(z0), 0) -> c5 PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) QUICKSORT(Cons(z0, Nil)) -> c11 QUICKSORT(Nil) -> c12 PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) APP(Nil, z0) -> c18 NOTEMPTY(Cons(z0, z1)) -> c19 NOTEMPTY(Nil) -> c20 PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) GOAL(z0) -> c23(QUICKSORT(z0)) S tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) QUICKSORT(Cons(z0, Nil)) -> c11 QUICKSORT(Nil) -> c12 PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) APP(Nil, z0) -> c18 NOTEMPTY(Cons(z0, z1)) -> c19 NOTEMPTY(Nil) -> c20 PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) GOAL(z0) -> c23(QUICKSORT(z0)) K tuples:none Defined Rule Symbols: quicksort_1, partLt_2, partGt_2, app_2, notEmpty_1, part_2, goal_1, <_2, >_2, partLt[Ite][True][Ite]_3, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTLT_2, PARTGT_2, APP_2, NOTEMPTY_1, PART_2, GOAL_1 Compound Symbols: c_1, c1, c2, c3_1, c4, c5, c6_1, c7_1, c8_1, c9_1, c10_1, c11, c12, c13_2, c14, c15_2, c16, c17_1, c18, c19, c20, c21_3, c22_3, c23_1 ---------------------------------------- (71) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: GOAL(z0) -> c23(QUICKSORT(z0)) Removed 9 trailing nodes: >'(0, z0) -> c4 NOTEMPTY(Cons(z0, z1)) -> c19 QUICKSORT(Cons(z0, Nil)) -> c11 >'(S(z0), 0) -> c5 APP(Nil, z0) -> c18 <'(z0, 0) -> c2 NOTEMPTY(Nil) -> c20 QUICKSORT(Nil) -> c12 <'(0, S(z0)) -> c1 ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) goal(z0) -> quicksort(z0) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) S tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) K tuples:none Defined Rule Symbols: quicksort_1, partLt_2, partGt_2, app_2, notEmpty_1, part_2, goal_1, <_2, >_2, partLt[Ite][True][Ite]_3, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTLT_2, PARTGT_2, APP_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c13_2, c14, c15_2, c16, c17_1, c21_3, c22_3 ---------------------------------------- (73) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False goal(z0) -> quicksort(z0) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) S tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) K tuples:none Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTLT_2, PARTGT_2, APP_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c13_2, c14, c15_2, c16, c17_1, c21_3, c22_3 ---------------------------------------- (75) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) We considered the (Usable) Rules: partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partLt(z0, Nil) -> Nil partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) And the Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(<(x_1, x_2)) = [1] + x_1 + x_2 POL(<'(x_1, x_2)) = 0 POL(>(x_1, x_2)) = [1] + x_1 + x_2 POL(>'(x_1, x_2)) = 0 POL(APP(x_1, x_2)) = 0 POL(Cons(x_1, x_2)) = [1] + x_2 POL(False) = 0 POL(Nil) = 0 POL(PART(x_1, x_2)) = x_2 POL(PARTGT(x_1, x_2)) = 0 POL(PARTGT[ITE][TRUE][ITE](x_1, x_2, x_3)) = 0 POL(PARTLT(x_1, x_2)) = 0 POL(PARTLT[ITE][TRUE][ITE](x_1, x_2, x_3)) = 0 POL(QUICKSORT(x_1)) = x_1 POL(S(x_1)) = [1] + x_1 POL(True) = 0 POL(app(x_1, x_2)) = [1] + x_2 POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c14) = 0 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c16) = 0 POL(c17(x_1)) = x_1 POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(part(x_1, x_2)) = [1] + x_1 + x_2 POL(partGt(x_1, x_2)) = x_2 POL(partGt[Ite][True][Ite](x_1, x_2, x_3)) = x_3 POL(partLt(x_1, x_2)) = x_2 POL(partLt[Ite][True][Ite](x_1, x_2, x_3)) = x_3 POL(quicksort(x_1)) = 0 ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) S tuples: PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTLT_2, PARTGT_2, APP_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c13_2, c14, c15_2, c16, c17_1, c21_3, c22_3 ---------------------------------------- (77) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) S tuples: PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTLT_2, PARTGT_2, APP_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c13_2, c14, c15_2, c16, c17_1, c21_3, c22_3 ---------------------------------------- (79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. PARTGT(z0, Nil) -> c16 We considered the (Usable) Rules: >(S(z0), 0) -> True partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False partLt(z0, Nil) -> Nil partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) And the Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(<(x_1, x_2)) = [1] + x_1 + x_2 POL(<'(x_1, x_2)) = 0 POL(>(x_1, x_2)) = [1] POL(>'(x_1, x_2)) = 0 POL(APP(x_1, x_2)) = 0 POL(Cons(x_1, x_2)) = [1] + x_2 POL(False) = [1] POL(Nil) = 0 POL(PART(x_1, x_2)) = [1] + x_2 POL(PARTGT(x_1, x_2)) = [1] POL(PARTGT[ITE][TRUE][ITE](x_1, x_2, x_3)) = x_1 POL(PARTLT(x_1, x_2)) = 0 POL(PARTLT[ITE][TRUE][ITE](x_1, x_2, x_3)) = 0 POL(QUICKSORT(x_1)) = x_1 POL(S(x_1)) = [1] + x_1 POL(True) = [1] POL(app(x_1, x_2)) = [1] + x_2 POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c14) = 0 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c16) = 0 POL(c17(x_1)) = x_1 POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(part(x_1, x_2)) = [1] + x_1 + x_2 POL(partGt(x_1, x_2)) = x_2 POL(partGt[Ite][True][Ite](x_1, x_2, x_3)) = x_3 POL(partLt(x_1, x_2)) = x_2 POL(partLt[Ite][True][Ite](x_1, x_2, x_3)) = x_3 POL(quicksort(x_1)) = 0 ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) S tuples: PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) PARTGT(z0, Nil) -> c16 Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTLT_2, PARTGT_2, APP_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c13_2, c14, c15_2, c16, c17_1, c21_3, c22_3 ---------------------------------------- (81) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. PARTLT(z0, Nil) -> c14 We considered the (Usable) Rules: partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partLt(z0, Nil) -> Nil partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) And the Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(<(x_1, x_2)) = [1] + x_1 + x_2 POL(<'(x_1, x_2)) = 0 POL(>(x_1, x_2)) = [1] + x_1 + x_2 POL(>'(x_1, x_2)) = 0 POL(APP(x_1, x_2)) = 0 POL(Cons(x_1, x_2)) = [1] + x_2 POL(False) = 0 POL(Nil) = 0 POL(PART(x_1, x_2)) = [1] + x_2 POL(PARTGT(x_1, x_2)) = 0 POL(PARTGT[ITE][TRUE][ITE](x_1, x_2, x_3)) = 0 POL(PARTLT(x_1, x_2)) = [1] POL(PARTLT[ITE][TRUE][ITE](x_1, x_2, x_3)) = [1] POL(QUICKSORT(x_1)) = x_1 POL(S(x_1)) = [1] + x_1 POL(True) = 0 POL(app(x_1, x_2)) = [1] + x_2 POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c14) = 0 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c16) = 0 POL(c17(x_1)) = x_1 POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(part(x_1, x_2)) = [1] + x_1 + x_2 POL(partGt(x_1, x_2)) = x_2 POL(partGt[Ite][True][Ite](x_1, x_2, x_3)) = x_3 POL(partLt(x_1, x_2)) = x_2 POL(partLt[Ite][True][Ite](x_1, x_2, x_3)) = x_3 POL(quicksort(x_1)) = 0 ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) S tuples: PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) PARTGT(z0, Nil) -> c16 PARTLT(z0, Nil) -> c14 Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTLT_2, PARTGT_2, APP_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c13_2, c14, c15_2, c16, c17_1, c21_3, c22_3 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) by PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)), <'(0, S(z0))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)), <'(z0, 0)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)), <'(0, S(z0))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)), <'(z0, 0)) S tuples: PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)), <'(0, S(z0))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)), <'(z0, 0)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) PARTGT(z0, Nil) -> c16 PARTLT(z0, Nil) -> c14 Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTLT_2, PARTGT_2, APP_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c14, c15_2, c16, c17_1, c21_3, c22_3, c13_2 ---------------------------------------- (85) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: PARTGT(z0, Nil) -> c16 PARTLT(z0, Nil) -> c14 ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)), <'(0, S(z0))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)), <'(z0, 0)) S tuples: PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)), <'(0, S(z0))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)), <'(z0, 0)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTGT_2, APP_2, PART_2, PARTLT_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c15_2, c17_1, c21_3, c22_3, c13_2 ---------------------------------------- (87) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) S tuples: PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTGT_2, APP_2, PART_2, PARTLT_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c15_2, c17_1, c21_3, c22_3, c13_2, c13_1 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) by PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)), >'(0, z0)) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)), >'(S(z0), 0)) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)), >'(0, z0)) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)), >'(S(z0), 0)) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)), >'(0, z0)) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)), >'(S(z0), 0)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PART_2, PARTLT_2, PARTGT_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c21_3, c22_3, c13_2, c13_1, c15_2 ---------------------------------------- (91) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PART_2, PARTLT_2, PARTGT_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c21_3, c22_3, c13_2, c13_1, c15_2, c15_1 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) by PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Nil) -> c21(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))), QUICKSORT(partLt(z0, Nil)), PARTLT(z0, Nil)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Nil) -> c21(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))), QUICKSORT(partLt(z0, Nil)), PARTLT(z0, Nil)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Nil) -> c21(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))), QUICKSORT(partLt(z0, Nil)), PARTLT(z0, Nil)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Nil) -> c21(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))), QUICKSORT(partLt(z0, Nil)), PARTLT(z0, Nil)) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PART_2, PARTLT_2, PARTGT_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c22_3, c13_2, c13_1, c15_2, c15_1, c21_3 ---------------------------------------- (95) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Nil) -> c21(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))), QUICKSORT(partLt(z0, Nil))) PART(z0, Nil) -> c21(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))), QUICKSORT(partLt(z0, Nil))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PART_2, PARTLT_2, PARTGT_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c22_3, c13_2, c13_1, c15_2, c15_1, c21_3, c21_2 ---------------------------------------- (97) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PART_2, PARTLT_2, PARTGT_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c22_3, c13_2, c13_1, c15_2, c15_1, c21_3, c1_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) by PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c22(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))), QUICKSORT(partGt(z0, Nil)), PARTGT(z0, Nil)) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c22(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))), QUICKSORT(partGt(z0, Nil)), PARTGT(z0, Nil)) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c22(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))), QUICKSORT(partGt(z0, Nil)), PARTGT(z0, Nil)) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c22(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))), QUICKSORT(partGt(z0, Nil)), PARTGT(z0, Nil)) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_2, c13_1, c15_2, c15_1, c21_3, c1_1, c22_3 ---------------------------------------- (101) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c22(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))), QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c22(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))), QUICKSORT(partGt(z0, Nil))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_2, c13_1, c15_2, c15_1, c21_3, c1_1, c22_3, c22_2 ---------------------------------------- (103) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_2, c13_1, c15_2, c15_1, c21_3, c1_1, c22_3, c2_1 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) by PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_2, c15_1, c21_3, c1_1, c22_3, c2_1, c13_2 ---------------------------------------- (107) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) We considered the (Usable) Rules: partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) partLt(z0, Nil) -> Nil partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) And the Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) 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(>(x_1, x_2)) = x_1 POL(>'(x_1, x_2)) = 0 POL(APP(x_1, x_2)) = 0 POL(Cons(x_1, x_2)) = [1] + x_2 POL(False) = 0 POL(Nil) = 0 POL(PART(x_1, x_2)) = [1] + x_2 POL(PARTGT(x_1, x_2)) = 0 POL(PARTGT[ITE][TRUE][ITE](x_1, x_2, x_3)) = 0 POL(PARTLT(x_1, x_2)) = [1] POL(PARTLT[ITE][TRUE][ITE](x_1, x_2, x_3)) = [1] POL(QUICKSORT(x_1)) = x_1 POL(S(x_1)) = 0 POL(True) = 0 POL(app(x_1, x_2)) = [1] + x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c15(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c17(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(part(x_1, x_2)) = [1] + x_1 + x_2 POL(partGt(x_1, x_2)) = x_2 POL(partGt[Ite][True][Ite](x_1, x_2, x_3)) = x_3 POL(partLt(x_1, x_2)) = x_2 POL(partLt[Ite][True][Ite](x_1, x_2, x_3)) = x_3 POL(quicksort(x_1)) = 0 ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_2, c15_1, c21_3, c1_1, c22_3, c2_1, c13_2 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) by PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c21_3, c1_1, c22_3, c2_1, c13_2, c15_2 ---------------------------------------- (111) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) We considered the (Usable) Rules: partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) partLt(z0, Nil) -> Nil partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) And the Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) 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(>(x_1, x_2)) = 0 POL(>'(x_1, x_2)) = 0 POL(APP(x_1, x_2)) = 0 POL(Cons(x_1, x_2)) = [1] + x_2 POL(False) = 0 POL(Nil) = 0 POL(PART(x_1, x_2)) = [1] + x_2 POL(PARTGT(x_1, x_2)) = [1] POL(PARTGT[ITE][TRUE][ITE](x_1, x_2, x_3)) = [1] POL(PARTLT(x_1, x_2)) = 0 POL(PARTLT[ITE][TRUE][ITE](x_1, x_2, x_3)) = 0 POL(QUICKSORT(x_1)) = x_1 POL(S(x_1)) = 0 POL(True) = 0 POL(app(x_1, x_2)) = [1] + x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c15(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c17(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(part(x_1, x_2)) = [1] + x_1 + x_2 POL(partGt(x_1, x_2)) = x_2 POL(partGt[Ite][True][Ite](x_1, x_2, x_3)) = x_3 POL(partLt(x_1, x_2)) = x_2 POL(partLt[Ite][True][Ite](x_1, x_2, x_3)) = x_3 POL(quicksort(x_1)) = 0 ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c21_3, c1_1, c22_3, c2_1, c13_2, c15_2 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) by PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c21_3, c1_1, c22_3, c2_1, c13_2, c15_2 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) by PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c1_1, c22_3, c2_1, c13_2, c15_2, c21_3 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) by PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c1_1, c22_3, c2_1, c13_2, c15_2, c21_3 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) by PART(z0, Nil) -> c1(QUICKSORT(Nil)) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c1(QUICKSORT(Nil)) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c1_1, c22_3, c2_1, c13_2, c15_2, c21_3 ---------------------------------------- (121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: PART(z0, Nil) -> c1(QUICKSORT(Nil)) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c1_1, c22_3, c2_1, c13_2, c15_2, c21_3 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) by PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(x0, Nil) -> c1(APP(Nil, Cons(x0, quicksort(partGt(x0, Nil))))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(x0, Nil) -> c1(APP(Nil, Cons(x0, quicksort(partGt(x0, Nil))))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c1_1, c22_3, c2_1, c13_2, c15_2, c21_3 ---------------------------------------- (125) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: PART(x0, Nil) -> c1(APP(Nil, Cons(x0, quicksort(partGt(x0, Nil))))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c1_1, c22_3, c2_1, c13_2, c15_2, c21_3 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) by PART(z0, Nil) -> c1(QUICKSORT(Nil)) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c1(QUICKSORT(Nil)) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c22_3, c2_1, c13_2, c15_2, c21_3, c1_1 ---------------------------------------- (129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: PART(z0, Nil) -> c1(QUICKSORT(Nil)) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c22_3, c2_1, c13_2, c15_2, c21_3, c1_1 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) by PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c22_3, c2_1, c13_2, c15_2, c21_3, c1_1 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) by PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c2_1, c13_2, c15_2, c21_3, c1_1, c22_3 ---------------------------------------- (135) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) by PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c2_1, c13_2, c15_2, c21_3, c1_1, c22_3 ---------------------------------------- (137) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) by PART(z0, Nil) -> c2(QUICKSORT(Nil)) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(Nil)) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c2_1, c13_2, c15_2, c21_3, c1_1, c22_3 ---------------------------------------- (139) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: PART(z0, Nil) -> c2(QUICKSORT(Nil)) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c2_1, c13_2, c15_2, c21_3, c1_1, c22_3 ---------------------------------------- (141) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) by PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(x0, Nil) -> c2(APP(Nil, Cons(x0, quicksort(partGt(x0, Nil))))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(x0, Nil) -> c2(APP(Nil, Cons(x0, quicksort(partGt(x0, Nil))))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c2_1, c13_2, c15_2, c21_3, c1_1, c22_3 ---------------------------------------- (143) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: PART(x0, Nil) -> c2(APP(Nil, Cons(x0, quicksort(partGt(x0, Nil))))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c2_1, c13_2, c15_2, c21_3, c1_1, c22_3 ---------------------------------------- (145) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) by PART(z0, Nil) -> c2(QUICKSORT(Nil)) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(z0, Nil) -> c2(QUICKSORT(Nil)) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1 ---------------------------------------- (147) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: PART(z0, Nil) -> c2(QUICKSORT(Nil)) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1 ---------------------------------------- (149) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) by PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2 Compound Symbols: c_1, c3_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1 ---------------------------------------- (151) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) by PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3 Compound Symbols: c_1, c3_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1 ---------------------------------------- (153) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) by PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3 Compound Symbols: c_1, c3_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1 ---------------------------------------- (155) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) by PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3 Compound Symbols: c_1, c3_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1 ---------------------------------------- (157) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) by PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3 Compound Symbols: c_1, c3_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1 ---------------------------------------- (159) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) by PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3 Compound Symbols: c_1, c3_1, c9_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) by PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3 Compound Symbols: c_1, c3_1, c9_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1 ---------------------------------------- (163) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) by PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: <'(S(z0), S(z1)) -> c(<'(z0, z1)) >'(S(z0), S(z1)) -> c3(>'(z0, z1)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: <'_2, >'_2, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3 Compound Symbols: c_1, c3_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1, c9_1 ---------------------------------------- (165) 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))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: >'(S(z0), S(z1)) -> c3(>'(z0, z1)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0)))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: >'_2, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2 Compound Symbols: c3_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1, c9_1, c_1 ---------------------------------------- (167) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: >'(S(z0), S(z1)) -> c3(>'(z0, z1)) QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: >'_2, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2 Compound Symbols: c3_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1, c9_1, c_1 ---------------------------------------- (169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace >'(S(z0), S(z1)) -> c3(>'(z0, z1)) by >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2 Compound Symbols: c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1, c9_1, c_1, c3_1 ---------------------------------------- (171) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2 Compound Symbols: c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1, c9_1, c_1, c3_1 ---------------------------------------- (173) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) by PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) S tuples: APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2 Compound Symbols: c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1, c9_1, c_1, c3_1 ---------------------------------------- (175) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) by APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) S tuples: PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2 Compound Symbols: c10_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1, c9_1, c_1, c3_1, c17_1 ---------------------------------------- (177) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) by PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) S tuples: PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2 Compound Symbols: c10_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1, c9_1, c_1, c3_1, c17_1 ---------------------------------------- (179) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) by PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1))) S tuples: PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2 Compound Symbols: c10_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1, c9_1, c_1, c3_1, c17_1 ---------------------------------------- (181) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) by PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1))) PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1))) S tuples: PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2 Compound Symbols: c10_1, c13_1, c15_1, c13_2, c15_2, c1_1, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1 ---------------------------------------- (183) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) by PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1))) PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil))) S tuples: PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2 Compound Symbols: c10_1, c13_1, c15_1, c13_2, c15_2, c1_1, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1 ---------------------------------------- (185) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) by PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1))) PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil))) S tuples: PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2 Compound Symbols: c10_1, c13_1, c15_1, c13_2, c15_2, c1_1, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1 ---------------------------------------- (187) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) by PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1))) PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil))) S tuples: PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2 Compound Symbols: c10_1, c13_1, c15_1, c13_2, c15_2, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1 ---------------------------------------- (189) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) by PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1))) PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil))) PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2))) S tuples: PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2 Compound Symbols: c10_1, c13_1, c15_1, c13_2, c15_2, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1 ---------------------------------------- (191) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) by PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1))) PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil))) PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1))) S tuples: PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2 Compound Symbols: c10_1, c13_1, c15_1, c13_2, c15_2, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1 ---------------------------------------- (193) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) by PART(0, Cons(S(z0), z1)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z0), 0), 0, Cons(S(z0), z1))), PARTGT(0, Cons(S(z0), z1))) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1))) PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil))) PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z0), 0), 0, Cons(S(z0), z1))), PARTGT(0, Cons(S(z0), z1))) S tuples: PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2 Compound Symbols: c10_1, c13_1, c15_1, c13_2, c15_2, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1 ---------------------------------------- (195) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) by PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1))) PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil))) PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z0), 0), 0, Cons(S(z0), z1))), PARTGT(0, Cons(S(z0), z1))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) S tuples: PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2 Compound Symbols: c10_1, c13_1, c15_1, c13_2, c15_2, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1 ---------------------------------------- (197) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) by PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1))) PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil))) PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z0), 0), 0, Cons(S(z0), z1))), PARTGT(0, Cons(S(z0), z1))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) S tuples: PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2 Compound Symbols: c10_1, c13_1, c15_1, c13_2, c15_2, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1 ---------------------------------------- (199) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) by PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1))) PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil))) PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z0), 0), 0, Cons(S(z0), z1))), PARTGT(0, Cons(S(z0), z1))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2))) S tuples: PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2 Compound Symbols: c10_1, c13_1, c15_1, c13_2, c15_2, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1 ---------------------------------------- (201) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) by PART(S(z0), Cons(0, z1)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, S(z0)), S(z0), Cons(0, z1))), PARTGT(S(z0), Cons(0, z1))) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1))) PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil))) PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z0), 0), 0, Cons(S(z0), z1))), PARTGT(0, Cons(S(z0), z1))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, S(z0)), S(z0), Cons(0, z1))), PARTGT(S(z0), Cons(0, z1))) S tuples: PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2 Compound Symbols: c10_1, c13_1, c15_1, c13_2, c15_2, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1 ---------------------------------------- (203) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) by PART(0, Cons(z0, z1)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(z0, 0), 0, Cons(z0, z1))), PARTGT(0, Cons(z0, z1))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1))) PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil))) PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z0), 0), 0, Cons(S(z0), z1))), PARTGT(0, Cons(S(z0), z1))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, S(z0)), S(z0), Cons(0, z1))), PARTGT(S(z0), Cons(0, z1))) PART(0, Cons(z0, z1)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(z0, 0), 0, Cons(z0, z1))), PARTGT(0, Cons(z0, z1))) S tuples: PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2 Compound Symbols: c10_1, c13_1, c15_1, c13_2, c15_2, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1, c22_3 ---------------------------------------- (205) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) by PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, Nil))) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1))) PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil))) PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z0), 0), 0, Cons(S(z0), z1))), PARTGT(0, Cons(S(z0), z1))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, S(z0)), S(z0), Cons(0, z1))), PARTGT(S(z0), Cons(0, z1))) PART(0, Cons(z0, z1)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(z0, 0), 0, Cons(z0, z1))), PARTGT(0, Cons(z0, z1))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, Nil))) S tuples: PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2 Compound Symbols: c10_1, c13_1, c15_1, c13_2, c15_2, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1, c22_3 ---------------------------------------- (207) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) by PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, Nil))) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) Tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1)) PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1)) PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2)) PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1))) PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1)) PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1)) PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1)) PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2)) PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1)) <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1))) PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1))) PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil))) PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2))) PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1))) PART(0, Cons(S(z0), z1)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z0), 0), 0, Cons(S(z0), z1))), PARTGT(0, Cons(S(z0), z1))) PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2))) PART(S(z0), Cons(0, z1)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, S(z0)), S(z0), Cons(0, z1))), PARTGT(S(z0), Cons(0, z1))) PART(0, Cons(z0, z1)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(z0, 0), 0, Cons(z0, z1))), PARTGT(0, Cons(z0, z1))) PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, Nil))) S tuples: PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2))) PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2))) PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2))) PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2))) PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1)))) PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1)))) PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2))) PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2))) PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2))) PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2))) APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2)) K tuples: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0))) PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0))) Defined Rule Symbols: <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3 Defined Pair Symbols: QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2 Compound Symbols: c10_1, c13_1, c15_1, c13_2, c15_2, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1, c22_3