KILLED proof of input_gqcjpsbM9e.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 10 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 271 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 110 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 267 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 78 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 234 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 105 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 487 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 208 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 2 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 1223 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 311 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 1130 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 266 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 1565 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 224 ms] (60) CpxRNTS (61) CompletionProof [UPPER BOUND(ID), 0 ms] (62) CpxTypedWeightedCompleteTrs (63) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (64) CpxRNTS (65) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (66) CdtProblem (67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 253 ms] (72) CdtProblem (73) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 89 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 2823 ms] (80) CdtProblem (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 3087 ms] (82) CdtProblem (83) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 2531 ms] (86) CdtProblem (87) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 2624 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRhsSimplificationProcessorProof [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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 8 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 66 ms] (134) CdtProblem (135) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 95 ms] (138) CdtProblem (139) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtRewritingProof [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) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtRewritingProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtForwardInstantiationProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtRhsSimplificationProcessorProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) app(nil, y) -> y app(add(n, x), y) -> add(n, app(x, y)) low(n, nil) -> nil low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x)) if_low(true, n, add(m, x)) -> add(m, low(n, x)) if_low(false, n, add(m, x)) -> low(n, x) high(n, nil) -> nil high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x)) if_high(true, n, add(m, x)) -> high(n, x) if_high(false, n, add(m, x)) -> add(m, high(n, x)) quicksort(nil) -> nil quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) quot(0', s(y)) -> 0' quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) app(nil, y) -> y app(add(n, x), y) -> add(n, app(x, y)) low(n, nil) -> nil low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x)) if_low(true, n, add(m, x)) -> add(m, low(n, x)) if_low(false, n, add(m, x)) -> low(n, x) high(n, nil) -> nil high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x)) if_high(true, n, add(m, x)) -> high(n, x) if_high(false, n, add(m, x)) -> add(m, high(n, x)) quicksort(nil) -> nil quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) app(nil, y) -> y app(add(n, x), y) -> add(n, app(x, y)) low(n, nil) -> nil low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x)) if_low(true, n, add(m, x)) -> add(m, low(n, x)) if_low(false, n, add(m, x)) -> low(n, x) high(n, nil) -> nil high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x)) if_high(true, n, add(m, x)) -> high(n, x) if_high(false, n, add(m, x)) -> add(m, high(n, x)) quicksort(nil) -> nil quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] quot(0, s(y)) -> 0 [1] quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] app(nil, y) -> y [1] app(add(n, x), y) -> add(n, app(x, y)) [1] low(n, nil) -> nil [1] low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x)) [1] if_low(true, n, add(m, x)) -> add(m, low(n, x)) [1] if_low(false, n, add(m, x)) -> low(n, x) [1] high(n, nil) -> nil [1] high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x)) [1] if_high(true, n, add(m, x)) -> high(n, x) [1] if_high(false, n, add(m, x)) -> add(m, high(n, x)) [1] quicksort(nil) -> nil [1] quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] quot(0, s(y)) -> 0 [1] quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] app(nil, y) -> y [1] app(add(n, x), y) -> add(n, app(x, y)) [1] low(n, nil) -> nil [1] low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x)) [1] if_low(true, n, add(m, x)) -> add(m, low(n, x)) [1] if_low(false, n, add(m, x)) -> low(n, x) [1] high(n, nil) -> nil [1] high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x)) [1] if_high(true, n, add(m, x)) -> high(n, x) [1] if_high(false, n, add(m, x)) -> add(m, high(n, x)) [1] quicksort(nil) -> nil [1] quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) [1] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s quot :: 0:s -> 0:s -> 0:s le :: 0:s -> 0:s -> true:false true :: true:false false :: true:false app :: nil:add -> nil:add -> nil:add nil :: nil:add add :: 0:s -> nil:add -> nil:add low :: 0:s -> nil:add -> nil:add if_low :: true:false -> 0:s -> nil:add -> nil:add high :: 0:s -> nil:add -> nil:add if_high :: true:false -> 0:s -> nil:add -> nil:add quicksort :: nil:add -> nil:add Rewrite Strategy: INNERMOST ---------------------------------------- (9) 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: quot_2 (c) The following functions are completely defined: minus_2 le_2 quicksort_1 low_2 high_2 if_high_3 if_low_3 app_2 Due to the following rules being added: minus(v0, v1) -> 0 [0] if_high(v0, v1, v2) -> nil [0] if_low(v0, v1, v2) -> nil [0] And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] quot(0, s(y)) -> 0 [1] quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] app(nil, y) -> y [1] app(add(n, x), y) -> add(n, app(x, y)) [1] low(n, nil) -> nil [1] low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x)) [1] if_low(true, n, add(m, x)) -> add(m, low(n, x)) [1] if_low(false, n, add(m, x)) -> low(n, x) [1] high(n, nil) -> nil [1] high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x)) [1] if_high(true, n, add(m, x)) -> high(n, x) [1] if_high(false, n, add(m, x)) -> add(m, high(n, x)) [1] quicksort(nil) -> nil [1] quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) [1] minus(v0, v1) -> 0 [0] if_high(v0, v1, v2) -> nil [0] if_low(v0, v1, v2) -> nil [0] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s quot :: 0:s -> 0:s -> 0:s le :: 0:s -> 0:s -> true:false true :: true:false false :: true:false app :: nil:add -> nil:add -> nil:add nil :: nil:add add :: 0:s -> nil:add -> nil:add low :: 0:s -> nil:add -> nil:add if_low :: true:false -> 0:s -> nil:add -> nil:add high :: 0:s -> nil:add -> nil:add if_high :: true:false -> 0:s -> nil:add -> nil:add quicksort :: nil:add -> nil:add Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] quot(0, s(y)) -> 0 [1] quot(s(x), s(0)) -> s(quot(x, s(0))) [2] quot(s(s(x')), s(s(y'))) -> s(quot(minus(x', y'), s(s(y')))) [2] quot(s(x), s(y)) -> s(quot(0, s(y))) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] app(nil, y) -> y [1] app(add(n, x), y) -> add(n, app(x, y)) [1] low(n, nil) -> nil [1] low(n, add(0, x)) -> if_low(true, n, add(0, x)) [2] low(0, add(s(x''), x)) -> if_low(false, 0, add(s(x''), x)) [2] low(s(y''), add(s(x1), x)) -> if_low(le(x1, y''), s(y''), add(s(x1), x)) [2] if_low(true, n, add(m, x)) -> add(m, low(n, x)) [1] if_low(false, n, add(m, x)) -> low(n, x) [1] high(n, nil) -> nil [1] high(n, add(0, x)) -> if_high(true, n, add(0, x)) [2] high(0, add(s(x2), x)) -> if_high(false, 0, add(s(x2), x)) [2] high(s(y1), add(s(x3), x)) -> if_high(le(x3, y1), s(y1), add(s(x3), x)) [2] if_high(true, n, add(m, x)) -> high(n, x) [1] if_high(false, n, add(m, x)) -> add(m, high(n, x)) [1] quicksort(nil) -> nil [1] quicksort(add(n, nil)) -> app(quicksort(nil), add(n, quicksort(nil))) [3] quicksort(add(n, add(m', x4))) -> app(quicksort(if_low(le(m', n), n, add(m', x4))), add(n, quicksort(if_high(le(m', n), n, add(m', x4))))) [3] minus(v0, v1) -> 0 [0] if_high(v0, v1, v2) -> nil [0] if_low(v0, v1, v2) -> nil [0] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s quot :: 0:s -> 0:s -> 0:s le :: 0:s -> 0:s -> true:false true :: true:false false :: true:false app :: nil:add -> nil:add -> nil:add nil :: nil:add add :: 0:s -> nil:add -> nil:add low :: 0:s -> nil:add -> nil:add if_low :: true:false -> 0:s -> nil:add -> nil:add high :: 0:s -> nil:add -> nil:add if_high :: true:false -> 0:s -> nil:add -> nil:add quicksort :: nil:add -> nil:add Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 nil => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y app(z, z') -{ 1 }-> 1 + n + app(x, y) :|: n >= 0, x >= 0, y >= 0, z = 1 + n + x, z' = y high(z, z') -{ 2 }-> if_high(le(x3, y1), 1 + y1, 1 + (1 + x3) + x) :|: z = 1 + y1, y1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 2 }-> if_high(1, n, 1 + 0 + x) :|: n >= 0, z = n, x >= 0, z' = 1 + 0 + x high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 if_high(z, z', z'') -{ 1 }-> high(n, x) :|: n >= 0, z = 1, z'' = 1 + m + x, x >= 0, z' = n, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(n, x) :|: n >= 0, z'' = 1 + m + x, x >= 0, z' = n, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(n, x) :|: n >= 0, z'' = 1 + m + x, x >= 0, z' = n, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(n, x) :|: n >= 0, z = 1, z'' = 1 + m + x, x >= 0, z' = n, m >= 0 le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 low(z, z') -{ 2 }-> if_low(le(x1, y''), 1 + y'', 1 + (1 + x1) + x) :|: z = 1 + y'', x1 >= 0, x >= 0, y'' >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, n, 1 + 0 + x) :|: n >= 0, z = n, x >= 0, z' = 1 + 0 + x low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 quicksort(z) -{ 3 }-> app(quicksort(if_low(le(m', n), n, 1 + m' + x4)), 1 + n + quicksort(if_high(le(m', n), n, 1 + m' + x4))) :|: n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + n + quicksort(0)) :|: z = 1 + n + 0, n >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 quot(z, z') -{ 2 }-> 1 + quot(x, 1 + 0) :|: x >= 0, z' = 1 + 0, z = 1 + x quot(z, z') -{ 2 }-> 1 + quot(minus(x', y'), 1 + (1 + y')) :|: z' = 1 + (1 + y'), x' >= 0, y' >= 0, z = 1 + (1 + x') quot(z, z') -{ 1 }-> 1 + quot(0, 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 2 }-> if_high(le(x3, z - 1), 1 + (z - 1), 1 + (1 + x3) + x) :|: z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 2 }-> if_low(le(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 3 }-> app(quicksort(if_low(le(m', n), n, 1 + m' + x4)), 1 + n + quicksort(if_high(le(m', n), n, 1 + m' + x4))) :|: n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 2 }-> 1 + quot(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { minus } { le } { app } { quot } { high, if_high } { low, if_low } { quicksort } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 2 }-> if_high(le(x3, z - 1), 1 + (z - 1), 1 + (1 + x3) + x) :|: z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 2 }-> if_low(le(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 3 }-> app(quicksort(if_low(le(m', n), n, 1 + m' + x4)), 1 + n + quicksort(if_high(le(m', n), n, 1 + m' + x4))) :|: n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 2 }-> 1 + quot(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 Function symbols to be analyzed: {minus}, {le}, {app}, {quot}, {high,if_high}, {low,if_low}, {quicksort} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 2 }-> if_high(le(x3, z - 1), 1 + (z - 1), 1 + (1 + x3) + x) :|: z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 2 }-> if_low(le(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 3 }-> app(quicksort(if_low(le(m', n), n, 1 + m' + x4)), 1 + n + quicksort(if_high(le(m', n), n, 1 + m' + x4))) :|: n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 2 }-> 1 + quot(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 Function symbols to be analyzed: {minus}, {le}, {app}, {quot}, {high,if_high}, {low,if_low}, {quicksort} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 2 }-> if_high(le(x3, z - 1), 1 + (z - 1), 1 + (1 + x3) + x) :|: z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 2 }-> if_low(le(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 3 }-> app(quicksort(if_low(le(m', n), n, 1 + m' + x4)), 1 + n + quicksort(if_high(le(m', n), n, 1 + m' + x4))) :|: n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 2 }-> 1 + quot(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 Function symbols to be analyzed: {minus}, {le}, {app}, {quot}, {high,if_high}, {low,if_low}, {quicksort} Previous analysis results are: minus: runtime: ?, size: O(n^1) [z] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 2 }-> if_high(le(x3, z - 1), 1 + (z - 1), 1 + (1 + x3) + x) :|: z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 2 }-> if_low(le(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 3 }-> app(quicksort(if_low(le(m', n), n, 1 + m' + x4)), 1 + n + quicksort(if_high(le(m', n), n, 1 + m' + x4))) :|: n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 2 }-> 1 + quot(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 Function symbols to be analyzed: {le}, {app}, {quot}, {high,if_high}, {low,if_low}, {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 2 }-> if_high(le(x3, z - 1), 1 + (z - 1), 1 + (1 + x3) + x) :|: z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 2 }-> if_low(le(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 3 }-> app(quicksort(if_low(le(m', n), n, 1 + m' + x4)), 1 + n + quicksort(if_high(le(m', n), n, 1 + m' + x4))) :|: n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 Function symbols to be analyzed: {le}, {app}, {quot}, {high,if_high}, {low,if_low}, {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: le 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 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 2 }-> if_high(le(x3, z - 1), 1 + (z - 1), 1 + (1 + x3) + x) :|: z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 2 }-> if_low(le(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 3 }-> app(quicksort(if_low(le(m', n), n, 1 + m' + x4)), 1 + n + quicksort(if_high(le(m', n), n, 1 + m' + x4))) :|: n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 Function symbols to be analyzed: {le}, {app}, {quot}, {high,if_high}, {low,if_low}, {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] le: runtime: ?, size: O(1) [1] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: le after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 2 }-> if_high(le(x3, z - 1), 1 + (z - 1), 1 + (1 + x3) + x) :|: z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 2 }-> if_low(le(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 3 }-> app(quicksort(if_low(le(m', n), n, 1 + m' + x4)), 1 + n + quicksort(if_high(le(m', n), n, 1 + m' + x4))) :|: n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 Function symbols to be analyzed: {app}, {quot}, {high,if_high}, {low,if_low}, {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 3 + z }-> if_high(s2, 1 + (z - 1), 1 + (1 + x3) + x) :|: s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 3 + z }-> if_low(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 7 + 2*n }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(if_high(s4, n, 1 + m' + x4))) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 Function symbols to be analyzed: {app}, {quot}, {high,if_high}, {low,if_low}, {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (33) 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' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 3 + z }-> if_high(s2, 1 + (z - 1), 1 + (1 + x3) + x) :|: s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 3 + z }-> if_low(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 7 + 2*n }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(if_high(s4, n, 1 + m' + x4))) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 Function symbols to be analyzed: {app}, {quot}, {high,if_high}, {low,if_low}, {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (35) 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 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 3 + z }-> if_high(s2, 1 + (z - 1), 1 + (1 + x3) + x) :|: s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 3 + z }-> if_low(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 7 + 2*n }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(if_high(s4, n, 1 + m' + x4))) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 Function symbols to be analyzed: {quot}, {high,if_high}, {low,if_low}, {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 3 + z }-> if_high(s2, 1 + (z - 1), 1 + (1 + x3) + x) :|: s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 3 + z }-> if_low(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 7 + 2*n }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(if_high(s4, n, 1 + m' + x4))) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 Function symbols to be analyzed: {quot}, {high,if_high}, {low,if_low}, {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: quot after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 3 + z }-> if_high(s2, 1 + (z - 1), 1 + (1 + x3) + x) :|: s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 3 + z }-> if_low(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 7 + 2*n }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(if_high(s4, n, 1 + m' + x4))) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 Function symbols to be analyzed: {quot}, {high,if_high}, {low,if_low}, {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] quot: runtime: ?, size: O(n^1) [z] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: quot after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 4 + 2*z + z*z' + z' ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 3 + z }-> if_high(s2, 1 + (z - 1), 1 + (1 + x3) + x) :|: s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 3 + z }-> if_low(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 7 + 2*n }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(if_high(s4, n, 1 + m' + x4))) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 Function symbols to be analyzed: {high,if_high}, {low,if_low}, {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [z] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 3 + z }-> if_high(s2, 1 + (z - 1), 1 + (1 + x3) + x) :|: s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 3 + z }-> if_low(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 7 + 2*n }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(if_high(s4, n, 1 + m' + x4))) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 4 + 3*z }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 5 + 2*s' + s'*z' + 2*z' }-> 1 + s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 5 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' - 1 >= 0 Function symbols to be analyzed: {high,if_high}, {low,if_low}, {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [z] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: high after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' Computed SIZE bound using KoAT for: if_high after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z'' ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 3 + z }-> if_high(s2, 1 + (z - 1), 1 + (1 + x3) + x) :|: s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 3 + z }-> if_low(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 7 + 2*n }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(if_high(s4, n, 1 + m' + x4))) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 4 + 3*z }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 5 + 2*s' + s'*z' + 2*z' }-> 1 + s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 5 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' - 1 >= 0 Function symbols to be analyzed: {high,if_high}, {low,if_low}, {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [z] high: runtime: ?, size: O(n^1) [z'] if_high: runtime: ?, size: O(n^1) [z''] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: high after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 11 + 3*z + z*z' + 4*z' Computed RUNTIME bound using KoAT for: if_high after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 24 + 6*z' + 2*z'*z'' + 8*z'' ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 3 + z }-> if_high(s2, 1 + (z - 1), 1 + (1 + x3) + x) :|: s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 3 + z }-> if_low(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 7 + 2*n }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(if_high(s4, n, 1 + m' + x4))) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 4 + 3*z }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 5 + 2*s' + s'*z' + 2*z' }-> 1 + s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 5 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' - 1 >= 0 Function symbols to be analyzed: {low,if_low}, {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [z] high: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z'] if_high: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z''] ---------------------------------------- (49) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 42 + 8*x + 8*x2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + x2) + x, z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 43 + 8*x + 2*x*z + 8*x3 + 2*x3*z + 11*z }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + x3) + x, s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s9 :|: s9 >= 0, s9 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s12 :|: s12 >= 0, s12 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> 1 + m + s13 :|: s13 >= 0, s13 <= x, z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 3 + z }-> if_low(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 39 + 8*m' + 2*m'*n + 10*n + 2*n*x4 + 8*x4 }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(s14)) :|: s14 >= 0, s14 <= 1 + m' + x4, s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 4 + 3*z }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 5 + 2*s' + s'*z' + 2*z' }-> 1 + s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 5 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' - 1 >= 0 Function symbols to be analyzed: {low,if_low}, {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [z] high: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z'] if_high: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z''] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: low after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' Computed SIZE bound using KoAT for: if_low 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 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 42 + 8*x + 8*x2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + x2) + x, z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 43 + 8*x + 2*x*z + 8*x3 + 2*x3*z + 11*z }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + x3) + x, s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s9 :|: s9 >= 0, s9 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s12 :|: s12 >= 0, s12 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> 1 + m + s13 :|: s13 >= 0, s13 <= x, z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 3 + z }-> if_low(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 39 + 8*m' + 2*m'*n + 10*n + 2*n*x4 + 8*x4 }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(s14)) :|: s14 >= 0, s14 <= 1 + m' + x4, s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 4 + 3*z }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 5 + 2*s' + s'*z' + 2*z' }-> 1 + s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 5 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' - 1 >= 0 Function symbols to be analyzed: {low,if_low}, {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [z] high: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z'] if_high: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z''] low: runtime: ?, size: O(n^1) [z'] if_low: runtime: ?, size: O(n^1) [z''] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: low after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 11 + 3*z + z*z' + 4*z' Computed RUNTIME bound using KoAT for: if_low after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 24 + 6*z' + 2*z'*z'' + 8*z'' ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 42 + 8*x + 8*x2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + x2) + x, z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 43 + 8*x + 2*x*z + 8*x3 + 2*x3*z + 11*z }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + x3) + x, s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s9 :|: s9 >= 0, s9 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s12 :|: s12 >= 0, s12 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> 1 + m + s13 :|: s13 >= 0, s13 <= x, z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 3 + z }-> if_low(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x'') + x) :|: x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 39 + 8*m' + 2*m'*n + 10*n + 2*n*x4 + 8*x4 }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(s14)) :|: s14 >= 0, s14 <= 1 + m' + x4, s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 4 + 3*z }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 5 + 2*s' + s'*z' + 2*z' }-> 1 + s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 5 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' - 1 >= 0 Function symbols to be analyzed: {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [z] high: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z'] if_high: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z''] low: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z'] if_low: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z''] ---------------------------------------- (55) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 42 + 8*x + 8*x2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + x2) + x, z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 43 + 8*x + 2*x*z + 8*x3 + 2*x3*z + 11*z }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + x3) + x, s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s9 :|: s9 >= 0, s9 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s12 :|: s12 >= 0, s12 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> 1 + m + s13 :|: s13 >= 0, s13 <= x, z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s19 :|: s19 >= 0, s19 <= x, z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> 1 + m + s18 :|: s18 >= 0, s18 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s15 :|: s15 >= 0, s15 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0 low(z, z') -{ 42 + 8*x + 8*x'' }-> s16 :|: s16 >= 0, s16 <= 1 + (1 + x'') + x, x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 43 + 8*x + 2*x*z + 8*x1 + 2*x1*z + 11*z }-> s17 :|: s17 >= 0, s17 <= 1 + (1 + x1) + x, s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 71 + 16*m' + 4*m'*n + 18*n + 4*n*x4 + 16*x4 }-> app(quicksort(s20), 1 + n + quicksort(s14)) :|: s20 >= 0, s20 <= 1 + m' + x4, s14 >= 0, s14 <= 1 + m' + x4, s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 4 + 3*z }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 5 + 2*s' + s'*z' + 2*z' }-> 1 + s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 5 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' - 1 >= 0 Function symbols to be analyzed: {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [z] high: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z'] if_high: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z''] low: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z'] if_low: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z''] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: quicksort after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 42 + 8*x + 8*x2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + x2) + x, z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 43 + 8*x + 2*x*z + 8*x3 + 2*x3*z + 11*z }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + x3) + x, s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s9 :|: s9 >= 0, s9 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s12 :|: s12 >= 0, s12 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> 1 + m + s13 :|: s13 >= 0, s13 <= x, z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s19 :|: s19 >= 0, s19 <= x, z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> 1 + m + s18 :|: s18 >= 0, s18 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s15 :|: s15 >= 0, s15 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0 low(z, z') -{ 42 + 8*x + 8*x'' }-> s16 :|: s16 >= 0, s16 <= 1 + (1 + x'') + x, x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 43 + 8*x + 2*x*z + 8*x1 + 2*x1*z + 11*z }-> s17 :|: s17 >= 0, s17 <= 1 + (1 + x1) + x, s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 71 + 16*m' + 4*m'*n + 18*n + 4*n*x4 + 16*x4 }-> app(quicksort(s20), 1 + n + quicksort(s14)) :|: s20 >= 0, s20 <= 1 + m' + x4, s14 >= 0, s14 <= 1 + m' + x4, s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 4 + 3*z }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 5 + 2*s' + s'*z' + 2*z' }-> 1 + s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 5 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' - 1 >= 0 Function symbols to be analyzed: {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [z] high: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z'] if_high: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z''] low: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z'] if_low: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z''] quicksort: runtime: ?, size: EXP ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: quicksort after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x high(z, z') -{ 42 + 8*x + 8*x2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + x2) + x, z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0 high(z, z') -{ 43 + 8*x + 2*x*z + 8*x3 + 2*x3*z + 11*z }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + x3) + x, s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0 high(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s9 :|: s9 >= 0, s9 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0 high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s12 :|: s12 >= 0, s12 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> 1 + m + s13 :|: s13 >= 0, s13 <= x, z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s19 :|: s19 >= 0, s19 <= x, z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_low(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> 1 + m + s18 :|: s18 >= 0, s18 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 low(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s15 :|: s15 >= 0, s15 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0 low(z, z') -{ 42 + 8*x + 8*x'' }-> s16 :|: s16 >= 0, s16 <= 1 + (1 + x'') + x, x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0 low(z, z') -{ 43 + 8*x + 2*x*z + 8*x1 + 2*x1*z + 11*z }-> s17 :|: s17 >= 0, s17 <= 1 + (1 + x1) + x, s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksort(z) -{ 71 + 16*m' + 4*m'*n + 18*n + 4*n*x4 + 16*x4 }-> app(quicksort(s20), 1 + n + quicksort(s14)) :|: s20 >= 0, s20 <= 1 + m' + x4, s14 >= 0, s14 <= 1 + m' + x4, s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4) quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 4 + 3*z }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 5 + 2*s' + s'*z' + 2*z' }-> 1 + s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0 quot(z, z') -{ 5 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' - 1 >= 0 Function symbols to be analyzed: {quicksort} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [z] high: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z'] if_high: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z''] low: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z'] if_low: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z''] quicksort: runtime: INF, size: EXP ---------------------------------------- (61) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: minus(v0, v1) -> null_minus [0] quot(v0, v1) -> null_quot [0] if_low(v0, v1, v2) -> null_if_low [0] if_high(v0, v1, v2) -> null_if_high [0] le(v0, v1) -> null_le [0] app(v0, v1) -> null_app [0] low(v0, v1) -> null_low [0] high(v0, v1) -> null_high [0] quicksort(v0) -> null_quicksort [0] And the following fresh constants: null_minus, null_quot, null_if_low, null_if_high, null_le, null_app, null_low, null_high, null_quicksort ---------------------------------------- (62) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] quot(0, s(y)) -> 0 [1] quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] app(nil, y) -> y [1] app(add(n, x), y) -> add(n, app(x, y)) [1] low(n, nil) -> nil [1] low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x)) [1] if_low(true, n, add(m, x)) -> add(m, low(n, x)) [1] if_low(false, n, add(m, x)) -> low(n, x) [1] high(n, nil) -> nil [1] high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x)) [1] if_high(true, n, add(m, x)) -> high(n, x) [1] if_high(false, n, add(m, x)) -> add(m, high(n, x)) [1] quicksort(nil) -> nil [1] quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) [1] minus(v0, v1) -> null_minus [0] quot(v0, v1) -> null_quot [0] if_low(v0, v1, v2) -> null_if_low [0] if_high(v0, v1, v2) -> null_if_high [0] le(v0, v1) -> null_le [0] app(v0, v1) -> null_app [0] low(v0, v1) -> null_low [0] high(v0, v1) -> null_high [0] quicksort(v0) -> null_quicksort [0] The TRS has the following type information: minus :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot 0 :: 0:s:null_minus:null_quot s :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot quot :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot le :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot -> true:false:null_le true :: true:false:null_le false :: true:false:null_le app :: nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort nil :: nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort add :: 0:s:null_minus:null_quot -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort low :: 0:s:null_minus:null_quot -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort if_low :: true:false:null_le -> 0:s:null_minus:null_quot -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort high :: 0:s:null_minus:null_quot -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort if_high :: true:false:null_le -> 0:s:null_minus:null_quot -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort quicksort :: nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort null_minus :: 0:s:null_minus:null_quot null_quot :: 0:s:null_minus:null_quot null_if_low :: nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort null_if_high :: nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort null_le :: true:false:null_le null_app :: nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort null_low :: nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort null_high :: nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort null_quicksort :: nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort Rewrite Strategy: INNERMOST ---------------------------------------- (63) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 2 false => 1 nil => 0 null_minus => 0 null_quot => 0 null_if_low => 0 null_if_high => 0 null_le => 0 null_app => 0 null_low => 0 null_high => 0 null_quicksort => 0 ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 app(z, z') -{ 1 }-> 1 + n + app(x, y) :|: n >= 0, x >= 0, y >= 0, z = 1 + n + x, z' = y high(z, z') -{ 1 }-> if_high(le(m, n), n, 1 + m + x) :|: n >= 0, z' = 1 + m + x, z = n, x >= 0, m >= 0 high(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 high(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 if_high(z, z', z'') -{ 1 }-> high(n, x) :|: z = 2, n >= 0, z'' = 1 + m + x, x >= 0, z' = n, m >= 0 if_high(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if_high(z, z', z'') -{ 1 }-> 1 + m + high(n, x) :|: n >= 0, z = 1, z'' = 1 + m + x, x >= 0, z' = n, m >= 0 if_low(z, z', z'') -{ 1 }-> low(n, x) :|: n >= 0, z = 1, z'' = 1 + m + x, x >= 0, z' = n, m >= 0 if_low(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if_low(z, z', z'') -{ 1 }-> 1 + m + low(n, x) :|: z = 2, n >= 0, z'' = 1 + m + x, x >= 0, z' = n, m >= 0 le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x le(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y le(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 low(z, z') -{ 1 }-> if_low(le(m, n), n, 1 + m + x) :|: n >= 0, z' = 1 + m + x, z = n, x >= 0, m >= 0 low(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 low(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 quicksort(z) -{ 1 }-> app(quicksort(low(n, x)), 1 + n + quicksort(high(n, x))) :|: n >= 0, x >= 0, z = 1 + n + x quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 quot(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 quot(z, z') -{ 1 }-> 1 + quot(minus(x, y), 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (65) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) Tuples: MINUS(z0, 0) -> c MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) QUOT(0, s(z0)) -> c2 QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) LE(0, z0) -> c4 LE(s(z0), 0) -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(nil, z0) -> c7 APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, nil) -> c9 LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, nil) -> c13 HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(nil) -> c17 QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) S tuples: MINUS(z0, 0) -> c MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) QUOT(0, s(z0)) -> c2 QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) LE(0, z0) -> c4 LE(s(z0), 0) -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(nil, z0) -> c7 APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, nil) -> c9 LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, nil) -> c13 HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(nil) -> c17 QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) K tuples:none Defined Rule Symbols: minus_2, quot_2, le_2, app_2, low_2, if_low_3, high_2, if_high_3, quicksort_1 Defined Pair Symbols: MINUS_2, QUOT_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1 Compound Symbols: c, c1_1, c2, c3_2, c4, c5, c6_1, c7, c8_1, c9, c10_2, c11_1, c12_1, c13, c14_2, c15_1, c16_1, c17, c18_3, c19_3 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 8 trailing nodes: APP(nil, z0) -> c7 QUOT(0, s(z0)) -> c2 MINUS(z0, 0) -> c LE(0, z0) -> c4 HIGH(z0, nil) -> c13 QUICKSORT(nil) -> c17 LOW(z0, nil) -> c9 LE(s(z0), 0) -> c5 ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) K tuples:none Defined Rule Symbols: minus_2, quot_2, le_2, app_2, low_2, if_low_3, high_2, if_high_3, quicksort_1 Defined Pair Symbols: MINUS_2, QUOT_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3 ---------------------------------------- (69) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) K tuples:none Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, QUOT_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3 ---------------------------------------- (71) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) We considered the (Usable) Rules: minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 And the Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(APP(x_1, x_2)) = 0 POL(HIGH(x_1, x_2)) = 0 POL(IF_HIGH(x_1, x_2, x_3)) = x_3 POL(IF_LOW(x_1, x_2, x_3)) = x_3 POL(LE(x_1, x_2)) = 0 POL(LOW(x_1, x_2)) = 0 POL(MINUS(x_1, x_2)) = 0 POL(QUICKSORT(x_1)) = 0 POL(QUOT(x_1, x_2)) = x_1 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = [1] + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c18(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c19(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(false) = 0 POL(high(x_1, x_2)) = x_1 + x_2 POL(if_high(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(if_low(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(le(x_1, x_2)) = x_1 + x_2 POL(low(x_1, x_2)) = x_1 + x_2 POL(minus(x_1, x_2)) = x_1 POL(nil) = [1] POL(quicksort(x_1)) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, QUOT_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3 ---------------------------------------- (73) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) We considered the (Usable) Rules: low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) low(z0, nil) -> nil if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) minus(s(z0), s(z1)) -> minus(z0, z1) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) minus(z0, 0) -> z0 if_low(false, z0, add(z1, z2)) -> low(z0, z2) if_high(true, z0, add(z1, z2)) -> high(z0, z2) And the Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(APP(x_1, x_2)) = 0 POL(HIGH(x_1, x_2)) = 0 POL(IF_HIGH(x_1, x_2, x_3)) = 0 POL(IF_LOW(x_1, x_2, x_3)) = 0 POL(LE(x_1, x_2)) = 0 POL(LOW(x_1, x_2)) = 0 POL(MINUS(x_1, x_2)) = 0 POL(QUICKSORT(x_1)) = x_1 POL(QUOT(x_1, x_2)) = x_1 POL(add(x_1, x_2)) = [1] + x_2 POL(app(x_1, x_2)) = [1] + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c18(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c19(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(false) = 0 POL(high(x_1, x_2)) = x_2 POL(if_high(x_1, x_2, x_3)) = x_3 POL(if_low(x_1, x_2, x_3)) = x_3 POL(le(x_1, x_2)) = x_1 + x_2 POL(low(x_1, x_2)) = x_2 POL(minus(x_1, x_2)) = [1] + x_1 POL(nil) = [1] POL(quicksort(x_1)) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, QUOT_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) by QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)), MINUS(z0, 0)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)), MINUS(z0, 0)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2 Compound Symbols: c1_1, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2 Compound Symbols: c1_1, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1 ---------------------------------------- (79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) We considered the (Usable) Rules: minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 And the Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(HIGH(x_1, x_2)) = 0 POL(IF_HIGH(x_1, x_2, x_3)) = 0 POL(IF_LOW(x_1, x_2, x_3)) = 0 POL(LE(x_1, x_2)) = 0 POL(LOW(x_1, x_2)) = 0 POL(MINUS(x_1, x_2)) = [2] + x_2 POL(QUICKSORT(x_1)) = 0 POL(QUOT(x_1, x_2)) = x_1*x_2 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = [1] POL(c1(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c18(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c19(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(false) = 0 POL(high(x_1, x_2)) = 0 POL(if_high(x_1, x_2, x_3)) = [1] + x_2 + x_2^2 POL(if_low(x_1, x_2, x_3)) = [1] + x_2 + x_2^2 POL(le(x_1, x_2)) = 0 POL(low(x_1, x_2)) = 0 POL(minus(x_1, x_2)) = [1] + x_1 POL(nil) = 0 POL(quicksort(x_1)) = 0 POL(s(x_1)) = [2] + x_1 POL(true) = 0 ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) S tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2 Compound Symbols: c1_1, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1 ---------------------------------------- (81) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) We considered the (Usable) Rules: low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) low(z0, nil) -> nil if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) if_high(true, z0, add(z1, z2)) -> high(z0, z2) And the Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(HIGH(x_1, x_2)) = x_2 POL(IF_HIGH(x_1, x_2, x_3)) = x_3 POL(IF_LOW(x_1, x_2, x_3)) = 0 POL(LE(x_1, x_2)) = 0 POL(LOW(x_1, x_2)) = 0 POL(MINUS(x_1, x_2)) = 0 POL(QUICKSORT(x_1)) = [2]x_1^2 POL(QUOT(x_1, x_2)) = 0 POL(add(x_1, x_2)) = [1] + x_2 POL(app(x_1, x_2)) = [2] POL(c1(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c18(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c19(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(false) = 0 POL(high(x_1, x_2)) = x_2 POL(if_high(x_1, x_2, x_3)) = x_3 POL(if_low(x_1, x_2, x_3)) = x_3 POL(le(x_1, x_2)) = 0 POL(low(x_1, x_2)) = x_2 POL(minus(x_1, x_2)) = 0 POL(nil) = 0 POL(quicksort(x_1)) = 0 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) S tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2 Compound Symbols: c1_1, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1 ---------------------------------------- (83) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) S tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2 Compound Symbols: c1_1, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1 ---------------------------------------- (85) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) We considered the (Usable) Rules: low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) low(z0, nil) -> nil if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) if_high(true, z0, add(z1, z2)) -> high(z0, z2) And the Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(HIGH(x_1, x_2)) = 0 POL(IF_HIGH(x_1, x_2, x_3)) = 0 POL(IF_LOW(x_1, x_2, x_3)) = x_3 POL(LE(x_1, x_2)) = 0 POL(LOW(x_1, x_2)) = x_2 POL(MINUS(x_1, x_2)) = 0 POL(QUICKSORT(x_1)) = x_1^2 POL(QUOT(x_1, x_2)) = 0 POL(add(x_1, x_2)) = [2] + x_2 POL(app(x_1, x_2)) = [2] POL(c1(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c18(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c19(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(false) = 0 POL(high(x_1, x_2)) = x_2 POL(if_high(x_1, x_2, x_3)) = x_3 POL(if_low(x_1, x_2, x_3)) = x_3 POL(le(x_1, x_2)) = 0 POL(low(x_1, x_2)) = x_2 POL(minus(x_1, x_2)) = 0 POL(nil) = 0 POL(quicksort(x_1)) = 0 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) S tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2 Compound Symbols: c1_1, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1 ---------------------------------------- (87) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) S tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2 Compound Symbols: c1_1, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1 ---------------------------------------- (89) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. LE(s(z0), s(z1)) -> c6(LE(z0, z1)) We considered the (Usable) Rules: low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) low(z0, nil) -> nil if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) le(s(z0), s(z1)) -> le(z0, z1) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) le(s(z0), 0) -> false le(0, z0) -> true high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) if_high(true, z0, add(z1, z2)) -> high(z0, z2) And the Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(HIGH(x_1, x_2)) = [2]x_1 + [2]x_1*x_2 POL(IF_HIGH(x_1, x_2, x_3)) = [2]x_2*x_3 POL(IF_LOW(x_1, x_2, x_3)) = x_2*x_3 POL(LE(x_1, x_2)) = [2]x_2 POL(LOW(x_1, x_2)) = [2]x_1 + x_1*x_2 POL(MINUS(x_1, x_2)) = 0 POL(QUICKSORT(x_1)) = [2]x_1^2 POL(QUOT(x_1, x_2)) = 0 POL(add(x_1, x_2)) = [2] + x_1 + x_2 POL(app(x_1, x_2)) = [2] POL(c1(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c18(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c19(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(false) = [1] POL(high(x_1, x_2)) = [2] + x_2 POL(if_high(x_1, x_2, x_3)) = [2] + x_1*x_3 POL(if_low(x_1, x_2, x_3)) = [2] + x_3 POL(le(x_1, x_2)) = [1] POL(low(x_1, x_2)) = [2] + x_2 POL(minus(x_1, x_2)) = 0 POL(nil) = 0 POL(quicksort(x_1)) = 0 POL(s(x_1)) = [2] + x_1 POL(true) = [1] ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2 Compound Symbols: c1_1, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) by LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)), LE(0, z0)) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)), LE(s(z0), 0)) LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)), LE(0, z0)) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)), LE(s(z0), 0)) LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2, LOW_2 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1, c10_2 ---------------------------------------- (93) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2, LOW_2 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1, c10_2, c10_1 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) by HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)), LE(0, z0)) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)), LE(s(z0), 0)) HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)), LE(0, z0)) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)), LE(s(z0), 0)) HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, QUOT_2, LOW_2, HIGH_2 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1, c10_2, c10_1, c14_2 ---------------------------------------- (97) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, QUOT_2, LOW_2, HIGH_2 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1, c10_2, c10_1, c14_2, c14_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) by QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)), LOW(z0, nil)) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil)), LOW(z0, nil)) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)), LOW(z0, nil)) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil)), LOW(z0, nil)) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, QUOT_2, LOW_2, HIGH_2 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c19_3, c3_2, c3_1, c10_2, c10_1, c14_2, c14_1, c18_3 ---------------------------------------- (101) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, QUOT_2, LOW_2, HIGH_2 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c19_3, c3_2, c3_1, c10_2, c10_1, c14_2, c14_1, c18_3, c18_2 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) by QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)), HIGH(z0, nil)) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil)), HIGH(z0, nil)) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)), HIGH(z0, nil)) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil)), HIGH(z0, nil)) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_2, c3_1, c10_2, c10_1, c14_2, c14_1, c18_3, c18_2, c19_3 ---------------------------------------- (105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_2, c3_1, c10_2, c10_1, c14_2, c14_1, c18_3, c18_2, c19_3, c19_2 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) by QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_2, c10_1, c14_2, c14_1, c18_3, c18_2, c19_3, c19_2, c3_2 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) by LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_2, c14_1, c18_3, c18_2, c19_3, c19_2, c3_2, c10_2 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) by HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c18_3, c18_2, c19_3, c19_2, c3_2, c10_2, c14_2 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) by QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c18_3, c18_2, c19_3, c19_2, c3_2, c10_2, c14_2 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) by QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c18_2, c19_3, c19_2, c3_2, c10_2, c14_2, c18_3 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) by QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c18_2, c19_3, c19_2, c3_2, c10_2, c14_2, c18_3 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) by QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(x0, nil)) -> c18(APP(nil, add(x0, quicksort(high(x0, nil)))), QUICKSORT(low(x0, nil))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(x0, nil)) -> c18(APP(nil, add(x0, quicksort(high(x0, nil)))), QUICKSORT(low(x0, nil))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c19_3, c19_2, c3_2, c10_2, c14_2, c18_3, c18_2 ---------------------------------------- (121) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(x0, nil)) -> c18(QUICKSORT(low(x0, nil))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c19_3, c19_2, c3_2, c10_2, c14_2, c18_3, c18_2, c18_1 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) by QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(x0, nil)) -> c18(QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c19_3, c19_2, c3_2, c10_2, c14_2, c18_3, c18_2, c18_1 ---------------------------------------- (125) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) by QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(x0, nil)) -> c18(QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c19_2, c3_2, c10_2, c14_2, c18_3, c18_2, c18_1, c19_3 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) by QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(x0, nil)) -> c18(QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c19_2, c3_2, c10_2, c14_2, c18_3, c18_2, c18_1, c19_3 ---------------------------------------- (129) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) by QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(add(x0, nil)) -> c19(APP(nil, add(x0, quicksort(high(x0, nil)))), QUICKSORT(high(x0, nil))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(x0, nil)) -> c18(QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(add(x0, nil)) -> c19(APP(nil, add(x0, quicksort(high(x0, nil)))), QUICKSORT(high(x0, nil))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c18_1, c19_3, c19_2 ---------------------------------------- (131) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(x0, nil)) -> c18(QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(add(x0, nil)) -> c19(QUICKSORT(high(x0, nil))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c18_1, c19_3, c19_2, c19_1 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(add(x0, nil)) -> c18(QUICKSORT(low(x0, nil))) by QUICKSORT(add(z0, nil)) -> c18(QUICKSORT(nil)) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(add(x0, nil)) -> c19(QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(QUICKSORT(nil)) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c19_1, c18_1 ---------------------------------------- (135) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: QUICKSORT(add(z0, nil)) -> c18(QUICKSORT(nil)) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(add(x0, nil)) -> c19(QUICKSORT(high(x0, nil))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c19_1 ---------------------------------------- (137) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(add(x0, nil)) -> c19(QUICKSORT(high(x0, nil))) by QUICKSORT(add(z0, nil)) -> c19(QUICKSORT(nil)) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(add(z0, nil)) -> c19(QUICKSORT(nil)) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c19_1 ---------------------------------------- (139) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: QUICKSORT(add(z0, nil)) -> c19(QUICKSORT(nil)) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2 ---------------------------------------- (141) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) by IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1 ---------------------------------------- (143) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) by QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1 ---------------------------------------- (145) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) by QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) by QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c1_1, c6_1, c8_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1 ---------------------------------------- (149) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) by IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3 Compound Symbols: c1_1, c6_1, c8_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1 ---------------------------------------- (151) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) by IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3 Compound Symbols: c1_1, c6_1, c8_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1 ---------------------------------------- (153) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) by IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3 Compound Symbols: c1_1, c6_1, c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1 ---------------------------------------- (155) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) by QUOT(s(s(z0)), s(s(0))) -> c3(MINUS(s(z0), s(0))) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(0))) -> c3(MINUS(s(z0), s(0))) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3 Compound Symbols: c1_1, c6_1, c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1 ---------------------------------------- (157) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) by QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(0))) -> c3(MINUS(s(z0), s(0))) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: MINUS_2, LE_2, APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3 Compound Symbols: c1_1, c6_1, c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1 ---------------------------------------- (159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) by MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(0))) -> c3(MINUS(s(z0), s(0))) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: LE_2, APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2 Compound Symbols: c6_1, c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c1_1 ---------------------------------------- (161) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: QUOT(s(s(z0)), s(s(0))) -> c3(MINUS(s(z0), s(0))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: LE_2, APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2 Compound Symbols: c6_1, c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c1_1 ---------------------------------------- (163) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: LE_2, APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2 Compound Symbols: c6_1, c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c1_1 ---------------------------------------- (165) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) by QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: LE_2, APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2 Compound Symbols: c6_1, c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c1_1 ---------------------------------------- (167) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) by QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: LE_2, APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2 Compound Symbols: c6_1, c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c1_1 ---------------------------------------- (169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LE(s(z0), s(z1)) -> c6(LE(z0, z1)) by LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2 Compound Symbols: c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c1_1, c6_1 ---------------------------------------- (171) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2))) S tuples: APP(add(z0, z1), z2) -> c8(APP(z1, z2)) K tuples: IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2 Compound Symbols: c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c1_1, c6_1 ---------------------------------------- (173) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace APP(add(z0, z1), z2) -> c8(APP(z1, z2)) by APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2))) APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) S tuples: APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) K tuples: IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2 Compound Symbols: c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c1_1, c6_1, c8_1 ---------------------------------------- (175) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) by QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2))) APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) S tuples: APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) K tuples: IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2 Compound Symbols: c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c1_1, c6_1, c8_1 ---------------------------------------- (177) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) by QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2))) APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0))) S tuples: APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) K tuples: IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2 Compound Symbols: c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1 ---------------------------------------- (179) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) by QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2))) APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) S tuples: APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) K tuples: IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2 Compound Symbols: c10_1, c14_1, c3_2, c10_2, c14_2, c18_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1 ---------------------------------------- (181) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) by QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil)), QUICKSORT(nil)) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2))) APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil)), QUICKSORT(nil)) S tuples: APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) K tuples: IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2 Compound Symbols: c10_1, c14_1, c3_2, c10_2, c14_2, c18_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1 ---------------------------------------- (183) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2))) APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil))) S tuples: APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) K tuples: IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2 Compound Symbols: c10_1, c14_1, c3_2, c10_2, c14_2, c18_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1 ---------------------------------------- (185) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) by QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(nil)) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2))) APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(nil)) S tuples: APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) K tuples: IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2 Compound Symbols: c10_1, c14_1, c3_2, c10_2, c14_2, c18_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1 ---------------------------------------- (187) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2))) APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil)))) S tuples: APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) K tuples: IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2 Compound Symbols: c10_1, c14_1, c3_2, c10_2, c14_2, c18_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1 ---------------------------------------- (189) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) by QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(nil)) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2))) APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil)))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(nil)) S tuples: APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) K tuples: IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2 Compound Symbols: c10_1, c14_1, c3_2, c10_2, c14_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1, c18_2 ---------------------------------------- (191) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2))) APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil)))) S tuples: APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) K tuples: IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2 Compound Symbols: c10_1, c14_1, c3_2, c10_2, c14_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1 ---------------------------------------- (193) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) by QUICKSORT(add(z0, add(0, z1))) -> c19(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_high(le(0, z0), z0, add(0, z1))), HIGH(z0, add(0, z1))) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2))) APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil)))) QUICKSORT(add(z0, add(0, z1))) -> c19(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_high(le(0, z0), z0, add(0, z1))), HIGH(z0, add(0, z1))) S tuples: APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) K tuples: IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2 Compound Symbols: c10_1, c14_1, c3_2, c10_2, c14_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1 ---------------------------------------- (195) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) by QUICKSORT(add(0, add(s(z0), z1))) -> c19(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_high(le(s(z0), 0), 0, add(s(z0), z1))), HIGH(0, add(s(z0), z1))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2))) APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil)))) QUICKSORT(add(z0, add(0, z1))) -> c19(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_high(le(0, z0), z0, add(0, z1))), HIGH(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c19(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_high(le(s(z0), 0), 0, add(s(z0), z1))), HIGH(0, add(s(z0), z1))) S tuples: APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) K tuples: IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2 Compound Symbols: c10_1, c14_1, c3_2, c10_2, c14_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1 ---------------------------------------- (197) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) by QUICKSORT(add(s(z0), add(s(z1), z2))) -> c19(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_high(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), HIGH(s(z0), add(s(z1), z2))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2))) APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil)))) QUICKSORT(add(z0, add(0, z1))) -> c19(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_high(le(0, z0), z0, add(0, z1))), HIGH(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c19(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_high(le(s(z0), 0), 0, add(s(z0), z1))), HIGH(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c19(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_high(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), HIGH(s(z0), add(s(z1), z2))) S tuples: APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) K tuples: IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2 Compound Symbols: c10_1, c14_1, c3_2, c10_2, c14_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1 ---------------------------------------- (199) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) by QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_high(le(z1, z0), z0, add(z1, z2))), HIGH(z0, add(z1, z2))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2))) APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil)))) QUICKSORT(add(z0, add(0, z1))) -> c19(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_high(le(0, z0), z0, add(0, z1))), HIGH(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c19(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_high(le(s(z0), 0), 0, add(s(z0), z1))), HIGH(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c19(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_high(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), HIGH(s(z0), add(s(z1), z2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_high(le(z1, z0), z0, add(z1, z2))), HIGH(z0, add(z1, z2))) S tuples: APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) K tuples: IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2 Compound Symbols: c10_1, c14_1, c3_2, c10_2, c14_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1 ---------------------------------------- (201) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) by LOW(s(s(y1)), add(s(s(y0)), z2)) -> c10(LE(s(s(y0)), s(s(y1)))) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) quicksort(nil) -> nil quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))) low(z0, nil) -> nil low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2)) if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2)) if_low(false, z0, add(z1, z2)) -> low(z0, z2) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) high(z0, nil) -> nil high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2)) if_high(true, z0, add(z1, z2)) -> high(z0, z2) if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2)) Tuples: LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2))) LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2))) HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2))) HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), add(s(x1), x2)) -> c14(LE(s(x1), s(x0))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1)))) QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0)))) QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2))) LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2))) HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2))) HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2))) APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1))) QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil))) QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil)))) QUICKSORT(add(z0, add(0, z1))) -> c19(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_high(le(0, z0), z0, add(0, z1))), HIGH(z0, add(0, z1))) QUICKSORT(add(0, add(s(z0), z1))) -> c19(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_high(le(s(z0), 0), 0, add(s(z0), z1))), HIGH(0, add(s(z0), z1))) QUICKSORT(add(s(z0), add(s(z1), z2))) -> c19(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_high(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), HIGH(s(z0), add(s(z1), z2))) QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_high(le(z1, z0), z0, add(z1, z2))), HIGH(z0, add(z1, z2))) LOW(s(s(y1)), add(s(s(y0)), z2)) -> c10(LE(s(s(y0)), s(s(y1)))) S tuples: APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2)) K tuples: IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1)) IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1)) IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2)) IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1)) IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1)) IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2)) IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1)) IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1)) IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2)) IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1)) IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1)) IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2)) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) Defined Rule Symbols: minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3 Defined Pair Symbols: LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2 Compound Symbols: c10_1, c14_1, c3_2, c10_2, c14_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1