KILLED proof of input_RCCPL2QKsU.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 12 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 2 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 322 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 57 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 335 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 47 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 1477 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 457 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 373 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 104 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 1424 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 185 ms] (48) CpxRNTS (49) CompletionProof [UPPER BOUND(ID), 0 ms] (50) CpxTypedWeightedCompleteTrs (51) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (54) CdtProblem (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 211 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 3247 ms] (64) CdtProblem (65) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 3066 ms] (68) CdtProblem (69) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 3443 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 11 ms] (110) CdtProblem (111) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 121 ms] (114) CdtProblem (115) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 2 ms] (118) CdtProblem (119) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRewritingProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRewritingProof [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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtRewritingProof [BOTH BOUNDS(ID, ID), 2 ms] (178) CdtProblem (179) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (180) 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: le(0, Y) -> true le(s(X), 0) -> false le(s(X), s(Y)) -> le(X, Y) app(nil, Y) -> Y app(cons(N, L), Y) -> cons(N, app(L, Y)) low(N, nil) -> nil low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L)) iflow(true, N, cons(M, L)) -> cons(M, low(N, L)) iflow(false, N, cons(M, L)) -> low(N, L) high(N, nil) -> nil high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L)) ifhigh(true, N, cons(M, L)) -> high(N, L) ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L)) quicksort(nil) -> nil quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))) 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: le(0', Y) -> true le(s(X), 0') -> false le(s(X), s(Y)) -> le(X, Y) app(nil, Y) -> Y app(cons(N, L), Y) -> cons(N, app(L, Y)) low(N, nil) -> nil low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L)) iflow(true, N, cons(M, L)) -> cons(M, low(N, L)) iflow(false, N, cons(M, L)) -> low(N, L) high(N, nil) -> nil high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L)) ifhigh(true, N, cons(M, L)) -> high(N, L) ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L)) quicksort(nil) -> nil quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))) 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: le(0, Y) -> true le(s(X), 0) -> false le(s(X), s(Y)) -> le(X, Y) app(nil, Y) -> Y app(cons(N, L), Y) -> cons(N, app(L, Y)) low(N, nil) -> nil low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L)) iflow(true, N, cons(M, L)) -> cons(M, low(N, L)) iflow(false, N, cons(M, L)) -> low(N, L) high(N, nil) -> nil high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L)) ifhigh(true, N, cons(M, L)) -> high(N, L) ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L)) quicksort(nil) -> nil quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))) 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: 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(cons(N, L), Y) -> cons(N, app(L, Y)) [1] low(N, nil) -> nil [1] low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L)) [1] iflow(true, N, cons(M, L)) -> cons(M, low(N, L)) [1] iflow(false, N, cons(M, L)) -> low(N, L) [1] high(N, nil) -> nil [1] high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L)) [1] ifhigh(true, N, cons(M, L)) -> high(N, L) [1] ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L)) [1] quicksort(nil) -> nil [1] quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))) [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: 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(cons(N, L), Y) -> cons(N, app(L, Y)) [1] low(N, nil) -> nil [1] low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L)) [1] iflow(true, N, cons(M, L)) -> cons(M, low(N, L)) [1] iflow(false, N, cons(M, L)) -> low(N, L) [1] high(N, nil) -> nil [1] high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L)) [1] ifhigh(true, N, cons(M, L)) -> high(N, L) [1] ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L)) [1] quicksort(nil) -> nil [1] quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))) [1] The TRS has the following type information: le :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false app :: nil:cons -> nil:cons -> nil:cons nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons low :: 0:s -> nil:cons -> nil:cons iflow :: true:false -> 0:s -> nil:cons -> nil:cons high :: 0:s -> nil:cons -> nil:cons ifhigh :: true:false -> 0:s -> nil:cons -> nil:cons quicksort :: nil:cons -> nil:cons 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: none (c) The following functions are completely defined: le_2 quicksort_1 low_2 high_2 iflow_3 app_2 ifhigh_3 Due to the following rules being added: iflow(v0, v1, v2) -> nil [0] ifhigh(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: 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(cons(N, L), Y) -> cons(N, app(L, Y)) [1] low(N, nil) -> nil [1] low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L)) [1] iflow(true, N, cons(M, L)) -> cons(M, low(N, L)) [1] iflow(false, N, cons(M, L)) -> low(N, L) [1] high(N, nil) -> nil [1] high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L)) [1] ifhigh(true, N, cons(M, L)) -> high(N, L) [1] ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L)) [1] quicksort(nil) -> nil [1] quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))) [1] iflow(v0, v1, v2) -> nil [0] ifhigh(v0, v1, v2) -> nil [0] The TRS has the following type information: le :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false app :: nil:cons -> nil:cons -> nil:cons nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons low :: 0:s -> nil:cons -> nil:cons iflow :: true:false -> 0:s -> nil:cons -> nil:cons high :: 0:s -> nil:cons -> nil:cons ifhigh :: true:false -> 0:s -> nil:cons -> nil:cons quicksort :: nil:cons -> nil:cons 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: 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(cons(N, L), Y) -> cons(N, app(L, Y)) [1] low(N, nil) -> nil [1] low(N, cons(0, L)) -> iflow(true, N, cons(0, L)) [2] low(0, cons(s(X'), L)) -> iflow(false, 0, cons(s(X'), L)) [2] low(s(Y'), cons(s(X''), L)) -> iflow(le(X'', Y'), s(Y'), cons(s(X''), L)) [2] iflow(true, N, cons(M, L)) -> cons(M, low(N, L)) [1] iflow(false, N, cons(M, L)) -> low(N, L) [1] high(N, nil) -> nil [1] high(N, cons(0, L)) -> ifhigh(true, N, cons(0, L)) [2] high(0, cons(s(X1), L)) -> ifhigh(false, 0, cons(s(X1), L)) [2] high(s(Y''), cons(s(X2), L)) -> ifhigh(le(X2, Y''), s(Y''), cons(s(X2), L)) [2] ifhigh(true, N, cons(M, L)) -> high(N, L) [1] ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L)) [1] quicksort(nil) -> nil [1] quicksort(cons(N, nil)) -> app(quicksort(nil), cons(N, quicksort(nil))) [3] quicksort(cons(N, cons(M', L'))) -> app(quicksort(iflow(le(M', N), N, cons(M', L'))), cons(N, quicksort(ifhigh(le(M', N), N, cons(M', L'))))) [3] iflow(v0, v1, v2) -> nil [0] ifhigh(v0, v1, v2) -> nil [0] The TRS has the following type information: le :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false app :: nil:cons -> nil:cons -> nil:cons nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons low :: 0:s -> nil:cons -> nil:cons iflow :: true:false -> 0:s -> nil:cons -> nil:cons high :: 0:s -> nil:cons -> nil:cons ifhigh :: true:false -> 0:s -> nil:cons -> nil:cons quicksort :: nil:cons -> nil:cons 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 :|: z' = Y, Y >= 0, z = 0 app(z, z') -{ 1 }-> 1 + N + app(L, Y) :|: z' = Y, Y >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 2 }-> ifhigh(le(X2, Y''), 1 + Y'', 1 + (1 + X2) + L) :|: z = 1 + Y'', z' = 1 + (1 + X2) + L, Y'' >= 0, L >= 0, X2 >= 0 high(z, z') -{ 2 }-> ifhigh(1, N, 1 + 0 + L) :|: z = N, z' = 1 + 0 + L, L >= 0, N >= 0 high(z, z') -{ 2 }-> ifhigh(0, 0, 1 + (1 + X1) + L) :|: X1 >= 0, z' = 1 + (1 + X1) + L, L >= 0, z = 0 high(z, z') -{ 1 }-> 0 :|: z = N, z' = 0, N >= 0 ifhigh(z, z', z'') -{ 1 }-> high(N, L) :|: z = 1, z' = N, L >= 0, z'' = 1 + M + L, M >= 0, N >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 ifhigh(z, z', z'') -{ 1 }-> 1 + M + high(N, L) :|: z' = N, L >= 0, z = 0, z'' = 1 + M + L, M >= 0, N >= 0 iflow(z, z', z'') -{ 1 }-> low(N, L) :|: z' = N, L >= 0, z = 0, z'' = 1 + M + L, M >= 0, N >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 iflow(z, z', z'') -{ 1 }-> 1 + M + low(N, L) :|: z = 1, z' = N, L >= 0, z'' = 1 + M + L, M >= 0, N >= 0 le(z, z') -{ 1 }-> le(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 le(z, z') -{ 1 }-> 1 :|: z' = Y, Y >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z = 1 + X, X >= 0, z' = 0 low(z, z') -{ 2 }-> iflow(le(X'', Y'), 1 + Y', 1 + (1 + X'') + L) :|: z' = 1 + (1 + X'') + L, Y' >= 0, X'' >= 0, L >= 0, z = 1 + Y' low(z, z') -{ 2 }-> iflow(1, N, 1 + 0 + L) :|: z = N, z' = 1 + 0 + L, L >= 0, N >= 0 low(z, z') -{ 2 }-> iflow(0, 0, 1 + (1 + X') + L) :|: z' = 1 + (1 + X') + L, X' >= 0, L >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z = N, z' = 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(iflow(le(M', N), N, 1 + M' + L')), 1 + N + quicksort(ifhigh(le(M', N), N, 1 + M' + L'))) :|: L' >= 0, z = 1 + N + (1 + M' + L'), M' >= 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(0), 1 + N + quicksort(0)) :|: z = 1 + N + 0, N >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 ---------------------------------------- (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(L, z') :|: z' >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 2 }-> ifhigh(le(X2, z - 1), 1 + (z - 1), 1 + (1 + X2) + L) :|: z' = 1 + (1 + X2) + L, z - 1 >= 0, L >= 0, X2 >= 0 high(z, z') -{ 2 }-> ifhigh(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 high(z, z') -{ 2 }-> ifhigh(0, 0, 1 + (1 + X1) + L) :|: X1 >= 0, z' = 1 + (1 + X1) + L, L >= 0, z = 0 high(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 ifhigh(z, z', z'') -{ 1 }-> high(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ifhigh(z, z', z'') -{ 1 }-> 1 + M + high(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 1 }-> low(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 iflow(z, z', z'') -{ 1 }-> 1 + M + low(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 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 }-> iflow(le(X'', z - 1), 1 + (z - 1), 1 + (1 + X'') + L) :|: z' = 1 + (1 + X'') + L, z - 1 >= 0, X'' >= 0, L >= 0 low(z, z') -{ 2 }-> iflow(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 low(z, z') -{ 2 }-> iflow(0, 0, 1 + (1 + X') + L) :|: z' = 1 + (1 + X') + L, X' >= 0, L >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 quicksort(z) -{ 3 }-> app(quicksort(iflow(le(M', N), N, 1 + M' + L')), 1 + N + quicksort(ifhigh(le(M', N), N, 1 + M' + L'))) :|: L' >= 0, z = 1 + N + (1 + M' + L'), M' >= 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { le } { app } { ifhigh, high } { low, iflow } { 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(L, z') :|: z' >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 2 }-> ifhigh(le(X2, z - 1), 1 + (z - 1), 1 + (1 + X2) + L) :|: z' = 1 + (1 + X2) + L, z - 1 >= 0, L >= 0, X2 >= 0 high(z, z') -{ 2 }-> ifhigh(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 high(z, z') -{ 2 }-> ifhigh(0, 0, 1 + (1 + X1) + L) :|: X1 >= 0, z' = 1 + (1 + X1) + L, L >= 0, z = 0 high(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 ifhigh(z, z', z'') -{ 1 }-> high(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ifhigh(z, z', z'') -{ 1 }-> 1 + M + high(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 1 }-> low(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 iflow(z, z', z'') -{ 1 }-> 1 + M + low(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 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 }-> iflow(le(X'', z - 1), 1 + (z - 1), 1 + (1 + X'') + L) :|: z' = 1 + (1 + X'') + L, z - 1 >= 0, X'' >= 0, L >= 0 low(z, z') -{ 2 }-> iflow(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 low(z, z') -{ 2 }-> iflow(0, 0, 1 + (1 + X') + L) :|: z' = 1 + (1 + X') + L, X' >= 0, L >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 quicksort(z) -{ 3 }-> app(quicksort(iflow(le(M', N), N, 1 + M' + L')), 1 + N + quicksort(ifhigh(le(M', N), N, 1 + M' + L'))) :|: L' >= 0, z = 1 + N + (1 + M' + L'), M' >= 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {le}, {app}, {ifhigh,high}, {low,iflow}, {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(L, z') :|: z' >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 2 }-> ifhigh(le(X2, z - 1), 1 + (z - 1), 1 + (1 + X2) + L) :|: z' = 1 + (1 + X2) + L, z - 1 >= 0, L >= 0, X2 >= 0 high(z, z') -{ 2 }-> ifhigh(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 high(z, z') -{ 2 }-> ifhigh(0, 0, 1 + (1 + X1) + L) :|: X1 >= 0, z' = 1 + (1 + X1) + L, L >= 0, z = 0 high(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 ifhigh(z, z', z'') -{ 1 }-> high(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ifhigh(z, z', z'') -{ 1 }-> 1 + M + high(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 1 }-> low(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 iflow(z, z', z'') -{ 1 }-> 1 + M + low(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 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 }-> iflow(le(X'', z - 1), 1 + (z - 1), 1 + (1 + X'') + L) :|: z' = 1 + (1 + X'') + L, z - 1 >= 0, X'' >= 0, L >= 0 low(z, z') -{ 2 }-> iflow(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 low(z, z') -{ 2 }-> iflow(0, 0, 1 + (1 + X') + L) :|: z' = 1 + (1 + X') + L, X' >= 0, L >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 quicksort(z) -{ 3 }-> app(quicksort(iflow(le(M', N), N, 1 + M' + L')), 1 + N + quicksort(ifhigh(le(M', N), N, 1 + M' + L'))) :|: L' >= 0, z = 1 + N + (1 + M' + L'), M' >= 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {le}, {app}, {ifhigh,high}, {low,iflow}, {quicksort} ---------------------------------------- (21) 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 ---------------------------------------- (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(L, z') :|: z' >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 2 }-> ifhigh(le(X2, z - 1), 1 + (z - 1), 1 + (1 + X2) + L) :|: z' = 1 + (1 + X2) + L, z - 1 >= 0, L >= 0, X2 >= 0 high(z, z') -{ 2 }-> ifhigh(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 high(z, z') -{ 2 }-> ifhigh(0, 0, 1 + (1 + X1) + L) :|: X1 >= 0, z' = 1 + (1 + X1) + L, L >= 0, z = 0 high(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 ifhigh(z, z', z'') -{ 1 }-> high(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ifhigh(z, z', z'') -{ 1 }-> 1 + M + high(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 1 }-> low(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 iflow(z, z', z'') -{ 1 }-> 1 + M + low(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 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 }-> iflow(le(X'', z - 1), 1 + (z - 1), 1 + (1 + X'') + L) :|: z' = 1 + (1 + X'') + L, z - 1 >= 0, X'' >= 0, L >= 0 low(z, z') -{ 2 }-> iflow(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 low(z, z') -{ 2 }-> iflow(0, 0, 1 + (1 + X') + L) :|: z' = 1 + (1 + X') + L, X' >= 0, L >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 quicksort(z) -{ 3 }-> app(quicksort(iflow(le(M', N), N, 1 + M' + L')), 1 + N + quicksort(ifhigh(le(M', N), N, 1 + M' + L'))) :|: L' >= 0, z = 1 + N + (1 + M' + L'), M' >= 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {le}, {app}, {ifhigh,high}, {low,iflow}, {quicksort} Previous analysis results are: le: runtime: ?, size: O(1) [1] ---------------------------------------- (23) 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' ---------------------------------------- (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(L, z') :|: z' >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 2 }-> ifhigh(le(X2, z - 1), 1 + (z - 1), 1 + (1 + X2) + L) :|: z' = 1 + (1 + X2) + L, z - 1 >= 0, L >= 0, X2 >= 0 high(z, z') -{ 2 }-> ifhigh(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 high(z, z') -{ 2 }-> ifhigh(0, 0, 1 + (1 + X1) + L) :|: X1 >= 0, z' = 1 + (1 + X1) + L, L >= 0, z = 0 high(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 ifhigh(z, z', z'') -{ 1 }-> high(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ifhigh(z, z', z'') -{ 1 }-> 1 + M + high(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 1 }-> low(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 iflow(z, z', z'') -{ 1 }-> 1 + M + low(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 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 }-> iflow(le(X'', z - 1), 1 + (z - 1), 1 + (1 + X'') + L) :|: z' = 1 + (1 + X'') + L, z - 1 >= 0, X'' >= 0, L >= 0 low(z, z') -{ 2 }-> iflow(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 low(z, z') -{ 2 }-> iflow(0, 0, 1 + (1 + X') + L) :|: z' = 1 + (1 + X') + L, X' >= 0, L >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 quicksort(z) -{ 3 }-> app(quicksort(iflow(le(M', N), N, 1 + M' + L')), 1 + N + quicksort(ifhigh(le(M', N), N, 1 + M' + L'))) :|: L' >= 0, z = 1 + N + (1 + M' + L'), M' >= 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {app}, {ifhigh,high}, {low,iflow}, {quicksort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (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(L, z') :|: z' >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 3 + z }-> ifhigh(s'', 1 + (z - 1), 1 + (1 + X2) + L) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + X2) + L, z - 1 >= 0, L >= 0, X2 >= 0 high(z, z') -{ 2 }-> ifhigh(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 high(z, z') -{ 2 }-> ifhigh(0, 0, 1 + (1 + X1) + L) :|: X1 >= 0, z' = 1 + (1 + X1) + L, L >= 0, z = 0 high(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 ifhigh(z, z', z'') -{ 1 }-> high(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ifhigh(z, z', z'') -{ 1 }-> 1 + M + high(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 1 }-> low(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 iflow(z, z', z'') -{ 1 }-> 1 + M + low(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 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 }-> iflow(s', 1 + (z - 1), 1 + (1 + X'') + L) :|: s' >= 0, s' <= 1, z' = 1 + (1 + X'') + L, z - 1 >= 0, X'' >= 0, L >= 0 low(z, z') -{ 2 }-> iflow(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 low(z, z') -{ 2 }-> iflow(0, 0, 1 + (1 + X') + L) :|: z' = 1 + (1 + X') + L, X' >= 0, L >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 quicksort(z) -{ 7 + 2*N }-> app(quicksort(iflow(s1, N, 1 + M' + L')), 1 + N + quicksort(ifhigh(s2, N, 1 + M' + L'))) :|: s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, L' >= 0, z = 1 + N + (1 + M' + L'), M' >= 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {app}, {ifhigh,high}, {low,iflow}, {quicksort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (27) 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' ---------------------------------------- (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(L, z') :|: z' >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 3 + z }-> ifhigh(s'', 1 + (z - 1), 1 + (1 + X2) + L) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + X2) + L, z - 1 >= 0, L >= 0, X2 >= 0 high(z, z') -{ 2 }-> ifhigh(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 high(z, z') -{ 2 }-> ifhigh(0, 0, 1 + (1 + X1) + L) :|: X1 >= 0, z' = 1 + (1 + X1) + L, L >= 0, z = 0 high(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 ifhigh(z, z', z'') -{ 1 }-> high(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ifhigh(z, z', z'') -{ 1 }-> 1 + M + high(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 1 }-> low(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 iflow(z, z', z'') -{ 1 }-> 1 + M + low(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 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 }-> iflow(s', 1 + (z - 1), 1 + (1 + X'') + L) :|: s' >= 0, s' <= 1, z' = 1 + (1 + X'') + L, z - 1 >= 0, X'' >= 0, L >= 0 low(z, z') -{ 2 }-> iflow(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 low(z, z') -{ 2 }-> iflow(0, 0, 1 + (1 + X') + L) :|: z' = 1 + (1 + X') + L, X' >= 0, L >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 quicksort(z) -{ 7 + 2*N }-> app(quicksort(iflow(s1, N, 1 + M' + L')), 1 + N + quicksort(ifhigh(s2, N, 1 + M' + L'))) :|: s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, L' >= 0, z = 1 + N + (1 + M' + L'), M' >= 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {app}, {ifhigh,high}, {low,iflow}, {quicksort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (29) 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 ---------------------------------------- (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(L, z') :|: z' >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 3 + z }-> ifhigh(s'', 1 + (z - 1), 1 + (1 + X2) + L) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + X2) + L, z - 1 >= 0, L >= 0, X2 >= 0 high(z, z') -{ 2 }-> ifhigh(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 high(z, z') -{ 2 }-> ifhigh(0, 0, 1 + (1 + X1) + L) :|: X1 >= 0, z' = 1 + (1 + X1) + L, L >= 0, z = 0 high(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 ifhigh(z, z', z'') -{ 1 }-> high(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ifhigh(z, z', z'') -{ 1 }-> 1 + M + high(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 1 }-> low(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 iflow(z, z', z'') -{ 1 }-> 1 + M + low(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 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 }-> iflow(s', 1 + (z - 1), 1 + (1 + X'') + L) :|: s' >= 0, s' <= 1, z' = 1 + (1 + X'') + L, z - 1 >= 0, X'' >= 0, L >= 0 low(z, z') -{ 2 }-> iflow(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 low(z, z') -{ 2 }-> iflow(0, 0, 1 + (1 + X') + L) :|: z' = 1 + (1 + X') + L, X' >= 0, L >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 quicksort(z) -{ 7 + 2*N }-> app(quicksort(iflow(s1, N, 1 + M' + L')), 1 + N + quicksort(ifhigh(s2, N, 1 + M' + L'))) :|: s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, L' >= 0, z = 1 + N + (1 + M' + L'), M' >= 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {ifhigh,high}, {low,iflow}, {quicksort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (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') -{ 2 + L }-> 1 + N + s3 :|: s3 >= 0, s3 <= L + z', z' >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 3 + z }-> ifhigh(s'', 1 + (z - 1), 1 + (1 + X2) + L) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + X2) + L, z - 1 >= 0, L >= 0, X2 >= 0 high(z, z') -{ 2 }-> ifhigh(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 high(z, z') -{ 2 }-> ifhigh(0, 0, 1 + (1 + X1) + L) :|: X1 >= 0, z' = 1 + (1 + X1) + L, L >= 0, z = 0 high(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 ifhigh(z, z', z'') -{ 1 }-> high(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ifhigh(z, z', z'') -{ 1 }-> 1 + M + high(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 1 }-> low(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 iflow(z, z', z'') -{ 1 }-> 1 + M + low(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 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 }-> iflow(s', 1 + (z - 1), 1 + (1 + X'') + L) :|: s' >= 0, s' <= 1, z' = 1 + (1 + X'') + L, z - 1 >= 0, X'' >= 0, L >= 0 low(z, z') -{ 2 }-> iflow(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 low(z, z') -{ 2 }-> iflow(0, 0, 1 + (1 + X') + L) :|: z' = 1 + (1 + X') + L, X' >= 0, L >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 quicksort(z) -{ 7 + 2*N }-> app(quicksort(iflow(s1, N, 1 + M' + L')), 1 + N + quicksort(ifhigh(s2, N, 1 + M' + L'))) :|: s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, L' >= 0, z = 1 + N + (1 + M' + L'), M' >= 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {ifhigh,high}, {low,iflow}, {quicksort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: ifhigh after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z'' Computed SIZE bound using CoFloCo for: high after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + L }-> 1 + N + s3 :|: s3 >= 0, s3 <= L + z', z' >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 3 + z }-> ifhigh(s'', 1 + (z - 1), 1 + (1 + X2) + L) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + X2) + L, z - 1 >= 0, L >= 0, X2 >= 0 high(z, z') -{ 2 }-> ifhigh(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 high(z, z') -{ 2 }-> ifhigh(0, 0, 1 + (1 + X1) + L) :|: X1 >= 0, z' = 1 + (1 + X1) + L, L >= 0, z = 0 high(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 ifhigh(z, z', z'') -{ 1 }-> high(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ifhigh(z, z', z'') -{ 1 }-> 1 + M + high(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 1 }-> low(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 iflow(z, z', z'') -{ 1 }-> 1 + M + low(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 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 }-> iflow(s', 1 + (z - 1), 1 + (1 + X'') + L) :|: s' >= 0, s' <= 1, z' = 1 + (1 + X'') + L, z - 1 >= 0, X'' >= 0, L >= 0 low(z, z') -{ 2 }-> iflow(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 low(z, z') -{ 2 }-> iflow(0, 0, 1 + (1 + X') + L) :|: z' = 1 + (1 + X') + L, X' >= 0, L >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 quicksort(z) -{ 7 + 2*N }-> app(quicksort(iflow(s1, N, 1 + M' + L')), 1 + N + quicksort(ifhigh(s2, N, 1 + M' + L'))) :|: s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, L' >= 0, z = 1 + N + (1 + M' + L'), M' >= 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {ifhigh,high}, {low,iflow}, {quicksort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ifhigh: runtime: ?, size: O(n^1) [z''] high: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: ifhigh after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 10 + 2*z' + z'*z'' + 4*z'' Computed RUNTIME bound using KoAT for: high after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 54 + 7*z + 3*z*z' + 20*z' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + L }-> 1 + N + s3 :|: s3 >= 0, s3 <= L + z', z' >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 3 + z }-> ifhigh(s'', 1 + (z - 1), 1 + (1 + X2) + L) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + X2) + L, z - 1 >= 0, L >= 0, X2 >= 0 high(z, z') -{ 2 }-> ifhigh(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 high(z, z') -{ 2 }-> ifhigh(0, 0, 1 + (1 + X1) + L) :|: X1 >= 0, z' = 1 + (1 + X1) + L, L >= 0, z = 0 high(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 ifhigh(z, z', z'') -{ 1 }-> high(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ifhigh(z, z', z'') -{ 1 }-> 1 + M + high(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 1 }-> low(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 iflow(z, z', z'') -{ 1 }-> 1 + M + low(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 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 }-> iflow(s', 1 + (z - 1), 1 + (1 + X'') + L) :|: s' >= 0, s' <= 1, z' = 1 + (1 + X'') + L, z - 1 >= 0, X'' >= 0, L >= 0 low(z, z') -{ 2 }-> iflow(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 low(z, z') -{ 2 }-> iflow(0, 0, 1 + (1 + X') + L) :|: z' = 1 + (1 + X') + L, X' >= 0, L >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 quicksort(z) -{ 7 + 2*N }-> app(quicksort(iflow(s1, N, 1 + M' + L')), 1 + N + quicksort(ifhigh(s2, N, 1 + M' + L'))) :|: s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, L' >= 0, z = 1 + N + (1 + M' + L'), M' >= 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {low,iflow}, {quicksort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ifhigh: runtime: O(n^2) [10 + 2*z' + z'*z'' + 4*z''], size: O(n^1) [z''] high: runtime: O(n^2) [54 + 7*z + 3*z*z' + 20*z'], size: O(n^1) [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 + L }-> 1 + N + s3 :|: s3 >= 0, s3 <= L + z', z' >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 12 + 2*z + z*z' + 4*z' }-> s4 :|: s4 >= 0, s4 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0 high(z, z') -{ 20 + 4*L + 4*X1 }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + X1) + L, X1 >= 0, z' = 1 + (1 + X1) + L, L >= 0, z = 0 high(z, z') -{ 21 + 4*L + L*z + 4*X2 + X2*z + 5*z }-> s6 :|: s6 >= 0, s6 <= 1 + (1 + X2) + L, s'' >= 0, s'' <= 1, z' = 1 + (1 + X2) + L, z - 1 >= 0, L >= 0, X2 >= 0 high(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 ifhigh(z, z', z'') -{ 55 + 20*L + 3*L*z' + 7*z' }-> s7 :|: s7 >= 0, s7 <= L, z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ifhigh(z, z', z'') -{ 55 + 20*L + 3*L*z' + 7*z' }-> 1 + M + s8 :|: s8 >= 0, s8 <= L, L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 1 }-> low(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 iflow(z, z', z'') -{ 1 }-> 1 + M + low(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 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 }-> iflow(s', 1 + (z - 1), 1 + (1 + X'') + L) :|: s' >= 0, s' <= 1, z' = 1 + (1 + X'') + L, z - 1 >= 0, X'' >= 0, L >= 0 low(z, z') -{ 2 }-> iflow(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 low(z, z') -{ 2 }-> iflow(0, 0, 1 + (1 + X') + L) :|: z' = 1 + (1 + X') + L, X' >= 0, L >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 quicksort(z) -{ 21 + 4*L' + L'*N + 4*M' + M'*N + 5*N }-> app(quicksort(iflow(s1, N, 1 + M' + L')), 1 + N + quicksort(s9)) :|: s9 >= 0, s9 <= 1 + M' + L', s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, L' >= 0, z = 1 + N + (1 + M' + L'), M' >= 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {low,iflow}, {quicksort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ifhigh: runtime: O(n^2) [10 + 2*z' + z'*z'' + 4*z''], size: O(n^1) [z''] high: runtime: O(n^2) [54 + 7*z + 3*z*z' + 20*z'], size: O(n^1) [z'] ---------------------------------------- (39) 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: iflow 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 + L }-> 1 + N + s3 :|: s3 >= 0, s3 <= L + z', z' >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 12 + 2*z + z*z' + 4*z' }-> s4 :|: s4 >= 0, s4 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0 high(z, z') -{ 20 + 4*L + 4*X1 }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + X1) + L, X1 >= 0, z' = 1 + (1 + X1) + L, L >= 0, z = 0 high(z, z') -{ 21 + 4*L + L*z + 4*X2 + X2*z + 5*z }-> s6 :|: s6 >= 0, s6 <= 1 + (1 + X2) + L, s'' >= 0, s'' <= 1, z' = 1 + (1 + X2) + L, z - 1 >= 0, L >= 0, X2 >= 0 high(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 ifhigh(z, z', z'') -{ 55 + 20*L + 3*L*z' + 7*z' }-> s7 :|: s7 >= 0, s7 <= L, z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ifhigh(z, z', z'') -{ 55 + 20*L + 3*L*z' + 7*z' }-> 1 + M + s8 :|: s8 >= 0, s8 <= L, L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 1 }-> low(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 iflow(z, z', z'') -{ 1 }-> 1 + M + low(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 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 }-> iflow(s', 1 + (z - 1), 1 + (1 + X'') + L) :|: s' >= 0, s' <= 1, z' = 1 + (1 + X'') + L, z - 1 >= 0, X'' >= 0, L >= 0 low(z, z') -{ 2 }-> iflow(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 low(z, z') -{ 2 }-> iflow(0, 0, 1 + (1 + X') + L) :|: z' = 1 + (1 + X') + L, X' >= 0, L >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 quicksort(z) -{ 21 + 4*L' + L'*N + 4*M' + M'*N + 5*N }-> app(quicksort(iflow(s1, N, 1 + M' + L')), 1 + N + quicksort(s9)) :|: s9 >= 0, s9 <= 1 + M' + L', s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, L' >= 0, z = 1 + N + (1 + M' + L'), M' >= 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {low,iflow}, {quicksort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ifhigh: runtime: O(n^2) [10 + 2*z' + z'*z'' + 4*z''], size: O(n^1) [z''] high: runtime: O(n^2) [54 + 7*z + 3*z*z' + 20*z'], size: O(n^1) [z'] low: runtime: ?, size: O(n^1) [z'] iflow: runtime: ?, size: O(n^1) [z''] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: low after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1 + 2*z*z' + 18*z' Computed RUNTIME bound using KoAT for: iflow after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 4 + 4*z'*z'' + 36*z'' ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + L }-> 1 + N + s3 :|: s3 >= 0, s3 <= L + z', z' >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 12 + 2*z + z*z' + 4*z' }-> s4 :|: s4 >= 0, s4 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0 high(z, z') -{ 20 + 4*L + 4*X1 }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + X1) + L, X1 >= 0, z' = 1 + (1 + X1) + L, L >= 0, z = 0 high(z, z') -{ 21 + 4*L + L*z + 4*X2 + X2*z + 5*z }-> s6 :|: s6 >= 0, s6 <= 1 + (1 + X2) + L, s'' >= 0, s'' <= 1, z' = 1 + (1 + X2) + L, z - 1 >= 0, L >= 0, X2 >= 0 high(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 ifhigh(z, z', z'') -{ 55 + 20*L + 3*L*z' + 7*z' }-> s7 :|: s7 >= 0, s7 <= L, z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ifhigh(z, z', z'') -{ 55 + 20*L + 3*L*z' + 7*z' }-> 1 + M + s8 :|: s8 >= 0, s8 <= L, L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 1 }-> low(z', L) :|: L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 iflow(z, z', z'') -{ 1 }-> 1 + M + low(z', L) :|: z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 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 }-> iflow(s', 1 + (z - 1), 1 + (1 + X'') + L) :|: s' >= 0, s' <= 1, z' = 1 + (1 + X'') + L, z - 1 >= 0, X'' >= 0, L >= 0 low(z, z') -{ 2 }-> iflow(1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 low(z, z') -{ 2 }-> iflow(0, 0, 1 + (1 + X') + L) :|: z' = 1 + (1 + X') + L, X' >= 0, L >= 0, z = 0 low(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 quicksort(z) -{ 21 + 4*L' + L'*N + 4*M' + M'*N + 5*N }-> app(quicksort(iflow(s1, N, 1 + M' + L')), 1 + N + quicksort(s9)) :|: s9 >= 0, s9 <= 1 + M' + L', s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, L' >= 0, z = 1 + N + (1 + M' + L'), M' >= 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {quicksort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ifhigh: runtime: O(n^2) [10 + 2*z' + z'*z'' + 4*z''], size: O(n^1) [z''] high: runtime: O(n^2) [54 + 7*z + 3*z*z' + 20*z'], size: O(n^1) [z'] low: runtime: O(n^2) [1 + 2*z*z' + 18*z'], size: O(n^1) [z'] iflow: runtime: O(n^2) [4 + 4*z'*z'' + 36*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 + L }-> 1 + N + s3 :|: s3 >= 0, s3 <= L + z', z' >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 12 + 2*z + z*z' + 4*z' }-> s4 :|: s4 >= 0, s4 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0 high(z, z') -{ 20 + 4*L + 4*X1 }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + X1) + L, X1 >= 0, z' = 1 + (1 + X1) + L, L >= 0, z = 0 high(z, z') -{ 21 + 4*L + L*z + 4*X2 + X2*z + 5*z }-> s6 :|: s6 >= 0, s6 <= 1 + (1 + X2) + L, s'' >= 0, s'' <= 1, z' = 1 + (1 + X2) + L, z - 1 >= 0, L >= 0, X2 >= 0 high(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 ifhigh(z, z', z'') -{ 55 + 20*L + 3*L*z' + 7*z' }-> s7 :|: s7 >= 0, s7 <= L, z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ifhigh(z, z', z'') -{ 55 + 20*L + 3*L*z' + 7*z' }-> 1 + M + s8 :|: s8 >= 0, s8 <= L, L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 2 + 18*L + 2*L*z' }-> s14 :|: s14 >= 0, s14 <= L, L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 iflow(z, z', z'') -{ 2 + 18*L + 2*L*z' }-> 1 + M + s13 :|: s13 >= 0, s13 <= L, z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 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') -{ 6 + 4*z*z' + 36*z' }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0 low(z, z') -{ 78 + 36*L + 36*X' }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + X') + L, z' = 1 + (1 + X') + L, X' >= 0, L >= 0, z = 0 low(z, z') -{ 79 + 36*L + 4*L*z + 36*X'' + 4*X''*z + 9*z }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + X'') + L, s' >= 0, s' <= 1, z' = 1 + (1 + X'') + L, z - 1 >= 0, X'' >= 0, L >= 0 low(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 quicksort(z) -{ 61 + 40*L' + 5*L'*N + 40*M' + 5*M'*N + 9*N }-> app(quicksort(s15), 1 + N + quicksort(s9)) :|: s15 >= 0, s15 <= 1 + M' + L', s9 >= 0, s9 <= 1 + M' + L', s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, L' >= 0, z = 1 + N + (1 + M' + L'), M' >= 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {quicksort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ifhigh: runtime: O(n^2) [10 + 2*z' + z'*z'' + 4*z''], size: O(n^1) [z''] high: runtime: O(n^2) [54 + 7*z + 3*z*z' + 20*z'], size: O(n^1) [z'] low: runtime: O(n^2) [1 + 2*z*z' + 18*z'], size: O(n^1) [z'] iflow: runtime: O(n^2) [4 + 4*z'*z'' + 36*z''], size: O(n^1) [z''] ---------------------------------------- (45) 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: ? ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + L }-> 1 + N + s3 :|: s3 >= 0, s3 <= L + z', z' >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 12 + 2*z + z*z' + 4*z' }-> s4 :|: s4 >= 0, s4 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0 high(z, z') -{ 20 + 4*L + 4*X1 }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + X1) + L, X1 >= 0, z' = 1 + (1 + X1) + L, L >= 0, z = 0 high(z, z') -{ 21 + 4*L + L*z + 4*X2 + X2*z + 5*z }-> s6 :|: s6 >= 0, s6 <= 1 + (1 + X2) + L, s'' >= 0, s'' <= 1, z' = 1 + (1 + X2) + L, z - 1 >= 0, L >= 0, X2 >= 0 high(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 ifhigh(z, z', z'') -{ 55 + 20*L + 3*L*z' + 7*z' }-> s7 :|: s7 >= 0, s7 <= L, z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ifhigh(z, z', z'') -{ 55 + 20*L + 3*L*z' + 7*z' }-> 1 + M + s8 :|: s8 >= 0, s8 <= L, L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 2 + 18*L + 2*L*z' }-> s14 :|: s14 >= 0, s14 <= L, L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 iflow(z, z', z'') -{ 2 + 18*L + 2*L*z' }-> 1 + M + s13 :|: s13 >= 0, s13 <= L, z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 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') -{ 6 + 4*z*z' + 36*z' }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0 low(z, z') -{ 78 + 36*L + 36*X' }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + X') + L, z' = 1 + (1 + X') + L, X' >= 0, L >= 0, z = 0 low(z, z') -{ 79 + 36*L + 4*L*z + 36*X'' + 4*X''*z + 9*z }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + X'') + L, s' >= 0, s' <= 1, z' = 1 + (1 + X'') + L, z - 1 >= 0, X'' >= 0, L >= 0 low(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 quicksort(z) -{ 61 + 40*L' + 5*L'*N + 40*M' + 5*M'*N + 9*N }-> app(quicksort(s15), 1 + N + quicksort(s9)) :|: s15 >= 0, s15 <= 1 + M' + L', s9 >= 0, s9 <= 1 + M' + L', s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, L' >= 0, z = 1 + N + (1 + M' + L'), M' >= 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {quicksort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ifhigh: runtime: O(n^2) [10 + 2*z' + z'*z'' + 4*z''], size: O(n^1) [z''] high: runtime: O(n^2) [54 + 7*z + 3*z*z' + 20*z'], size: O(n^1) [z'] low: runtime: O(n^2) [1 + 2*z*z' + 18*z'], size: O(n^1) [z'] iflow: runtime: O(n^2) [4 + 4*z'*z'' + 36*z''], size: O(n^1) [z''] quicksort: runtime: ?, size: EXP ---------------------------------------- (47) 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: ? ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + L }-> 1 + N + s3 :|: s3 >= 0, s3 <= L + z', z' >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 12 + 2*z + z*z' + 4*z' }-> s4 :|: s4 >= 0, s4 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0 high(z, z') -{ 20 + 4*L + 4*X1 }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + X1) + L, X1 >= 0, z' = 1 + (1 + X1) + L, L >= 0, z = 0 high(z, z') -{ 21 + 4*L + L*z + 4*X2 + X2*z + 5*z }-> s6 :|: s6 >= 0, s6 <= 1 + (1 + X2) + L, s'' >= 0, s'' <= 1, z' = 1 + (1 + X2) + L, z - 1 >= 0, L >= 0, X2 >= 0 high(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 ifhigh(z, z', z'') -{ 55 + 20*L + 3*L*z' + 7*z' }-> s7 :|: s7 >= 0, s7 <= L, z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ifhigh(z, z', z'') -{ 55 + 20*L + 3*L*z' + 7*z' }-> 1 + M + s8 :|: s8 >= 0, s8 <= L, L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 2 + 18*L + 2*L*z' }-> s14 :|: s14 >= 0, s14 <= L, L >= 0, z = 0, z'' = 1 + M + L, M >= 0, z' >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 iflow(z, z', z'') -{ 2 + 18*L + 2*L*z' }-> 1 + M + s13 :|: s13 >= 0, s13 <= L, z = 1, L >= 0, z'' = 1 + M + L, M >= 0, z' >= 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') -{ 6 + 4*z*z' + 36*z' }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0 low(z, z') -{ 78 + 36*L + 36*X' }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + X') + L, z' = 1 + (1 + X') + L, X' >= 0, L >= 0, z = 0 low(z, z') -{ 79 + 36*L + 4*L*z + 36*X'' + 4*X''*z + 9*z }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + X'') + L, s' >= 0, s' <= 1, z' = 1 + (1 + X'') + L, z - 1 >= 0, X'' >= 0, L >= 0 low(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 quicksort(z) -{ 61 + 40*L' + 5*L'*N + 40*M' + 5*M'*N + 9*N }-> app(quicksort(s15), 1 + N + quicksort(s9)) :|: s15 >= 0, s15 <= 1 + M' + L', s9 >= 0, s9 <= 1 + M' + L', s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, L' >= 0, z = 1 + N + (1 + M' + L'), M' >= 0, N >= 0 quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {quicksort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ifhigh: runtime: O(n^2) [10 + 2*z' + z'*z'' + 4*z''], size: O(n^1) [z''] high: runtime: O(n^2) [54 + 7*z + 3*z*z' + 20*z'], size: O(n^1) [z'] low: runtime: O(n^2) [1 + 2*z*z' + 18*z'], size: O(n^1) [z'] iflow: runtime: O(n^2) [4 + 4*z'*z'' + 36*z''], size: O(n^1) [z''] quicksort: runtime: INF, size: EXP ---------------------------------------- (49) 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: iflow(v0, v1, v2) -> null_iflow [0] ifhigh(v0, v1, v2) -> null_ifhigh [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_iflow, null_ifhigh, null_app, null_low, null_high, null_quicksort ---------------------------------------- (50) 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: 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(cons(N, L), Y) -> cons(N, app(L, Y)) [1] low(N, nil) -> nil [1] low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L)) [1] iflow(true, N, cons(M, L)) -> cons(M, low(N, L)) [1] iflow(false, N, cons(M, L)) -> low(N, L) [1] high(N, nil) -> nil [1] high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L)) [1] ifhigh(true, N, cons(M, L)) -> high(N, L) [1] ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L)) [1] quicksort(nil) -> nil [1] quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))) [1] iflow(v0, v1, v2) -> null_iflow [0] ifhigh(v0, v1, v2) -> null_ifhigh [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: le :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false app :: nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort -> nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort -> nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort nil :: nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort cons :: 0:s -> nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort -> nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort low :: 0:s -> nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort -> nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort iflow :: true:false -> 0:s -> nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort -> nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort high :: 0:s -> nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort -> nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort ifhigh :: true:false -> 0:s -> nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort -> nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort quicksort :: nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort -> nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort null_iflow :: nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort null_ifhigh :: nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort null_app :: nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort null_low :: nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort null_high :: nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort null_quicksort :: nil:cons:null_iflow:null_ifhigh:null_app:null_low:null_high:null_quicksort Rewrite Strategy: INNERMOST ---------------------------------------- (51) 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 null_iflow => 0 null_ifhigh => 0 null_app => 0 null_low => 0 null_high => 0 null_quicksort => 0 ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> Y :|: z' = Y, Y >= 0, z = 0 app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 app(z, z') -{ 1 }-> 1 + N + app(L, Y) :|: z' = Y, Y >= 0, z = 1 + N + L, L >= 0, N >= 0 high(z, z') -{ 1 }-> ifhigh(le(M, N), N, 1 + M + L) :|: z' = 1 + M + L, z = N, L >= 0, M >= 0, N >= 0 high(z, z') -{ 1 }-> 0 :|: z = N, z' = 0, N >= 0 high(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 ifhigh(z, z', z'') -{ 1 }-> high(N, L) :|: z = 1, z' = N, L >= 0, z'' = 1 + M + L, M >= 0, N >= 0 ifhigh(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 ifhigh(z, z', z'') -{ 1 }-> 1 + M + high(N, L) :|: z' = N, L >= 0, z = 0, z'' = 1 + M + L, M >= 0, N >= 0 iflow(z, z', z'') -{ 1 }-> low(N, L) :|: z' = N, L >= 0, z = 0, z'' = 1 + M + L, M >= 0, N >= 0 iflow(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 iflow(z, z', z'') -{ 1 }-> 1 + M + low(N, L) :|: z = 1, z' = N, L >= 0, z'' = 1 + M + L, M >= 0, N >= 0 le(z, z') -{ 1 }-> le(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 le(z, z') -{ 1 }-> 1 :|: z' = Y, Y >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z = 1 + X, X >= 0, z' = 0 low(z, z') -{ 1 }-> iflow(le(M, N), N, 1 + M + L) :|: z' = 1 + M + L, z = N, L >= 0, M >= 0, N >= 0 low(z, z') -{ 1 }-> 0 :|: z = N, z' = 0, N >= 0 low(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 quicksort(z) -{ 1 }-> app(quicksort(low(N, L)), 1 + N + quicksort(high(N, L))) :|: z = 1 + N + L, L >= 0, N >= 0 quicksort(z) -{ 1 }-> 0 :|: z = 0 quicksort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (53) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(0, z0) -> c LE(s(z0), 0) -> c1 LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(nil, z0) -> c3 APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) LOW(z0, nil) -> c5 LOW(z0, cons(z1, z2)) -> c6(IFLOW(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, nil) -> c9 HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(nil) -> c13 QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) S tuples: LE(0, z0) -> c LE(s(z0), 0) -> c1 LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(nil, z0) -> c3 APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) LOW(z0, nil) -> c5 LOW(z0, cons(z1, z2)) -> c6(IFLOW(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, nil) -> c9 HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(nil) -> c13 QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) K tuples:none Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, LOW_2, IFLOW_3, HIGH_2, IFHIGH_3, QUICKSORT_1 Compound Symbols: c, c1, c2_1, c3, c4_1, c5, c6_2, c7_1, c8_1, c9, c10_2, c11_1, c12_1, c13, c14_3, c15_3 ---------------------------------------- (55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing nodes: LOW(z0, nil) -> c5 APP(nil, z0) -> c3 LE(s(z0), 0) -> c1 QUICKSORT(nil) -> c13 HIGH(z0, nil) -> c9 LE(0, z0) -> c ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) LOW(z0, cons(z1, z2)) -> c6(IFLOW(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) LOW(z0, cons(z1, z2)) -> c6(IFLOW(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) K tuples:none Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, LOW_2, IFLOW_3, HIGH_2, IFHIGH_3, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c6_2, c7_1, c8_1, c10_2, c11_1, c12_1, c14_3, c15_3 ---------------------------------------- (57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) We considered the (Usable) Rules: low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) high(z0, nil) -> nil ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) LOW(z0, cons(z1, z2)) -> c6(IFLOW(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(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(IFHIGH(x_1, x_2, x_3)) = 0 POL(IFLOW(x_1, x_2, x_3)) = 0 POL(LE(x_1, x_2)) = 0 POL(LOW(x_1, x_2)) = 0 POL(QUICKSORT(x_1)) = x_1 POL(app(x_1, x_2)) = [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_3)) = x_1 + x_2 + x_3 POL(c15(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(false) = 0 POL(high(x_1, x_2)) = x_2 POL(ifhigh(x_1, x_2, x_3)) = x_3 POL(iflow(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(nil) = [1] POL(quicksort(x_1)) = [1] POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) LOW(z0, cons(z1, z2)) -> c6(IFLOW(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) LOW(z0, cons(z1, z2)) -> c6(IFLOW(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, LOW_2, IFLOW_3, HIGH_2, IFHIGH_3, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c6_2, c7_1, c8_1, c10_2, c11_1, c12_1, c14_3, c15_3 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LOW(z0, cons(z1, z2)) -> c6(IFLOW(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) by LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2)), LE(0, z0)) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2)), LE(s(z0), 0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2)), LE(0, z0)) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2)), LE(s(z0), 0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2)), LE(0, z0)) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2)), LE(s(z0), 0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, HIGH_2, IFHIGH_3, QUICKSORT_1, LOW_2 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c10_2, c11_1, c12_1, c14_3, c15_3, c6_2 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, HIGH_2, IFHIGH_3, QUICKSORT_1, LOW_2 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c10_2, c11_1, c12_1, c14_3, c15_3, c6_2, c6_1 ---------------------------------------- (63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) We considered the (Usable) Rules: low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) high(z0, nil) -> nil ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) 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(IFHIGH(x_1, x_2, x_3)) = x_3 POL(IFLOW(x_1, x_2, x_3)) = 0 POL(LE(x_1, x_2)) = 0 POL(LOW(x_1, x_2)) = 0 POL(QUICKSORT(x_1)) = [2]x_1^2 POL(app(x_1, x_2)) = [2] 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_3)) = x_1 + x_2 + x_3 POL(c15(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(false) = 0 POL(high(x_1, x_2)) = x_2 POL(ifhigh(x_1, x_2, x_3)) = x_3 POL(iflow(x_1, x_2, x_3)) = x_3 POL(le(x_1, x_2)) = 0 POL(low(x_1, x_2)) = x_2 POL(nil) = 0 POL(quicksort(x_1)) = 0 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, HIGH_2, IFHIGH_3, QUICKSORT_1, LOW_2 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c10_2, c11_1, c12_1, c14_3, c15_3, c6_2, c6_1 ---------------------------------------- (65) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, HIGH_2, IFHIGH_3, QUICKSORT_1, LOW_2 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c10_2, c11_1, c12_1, c14_3, c15_3, c6_2, c6_1 ---------------------------------------- (67) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) We considered the (Usable) Rules: low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) le(0, z0) -> true high(z0, nil) -> nil ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) 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(IFHIGH(x_1, x_2, x_3)) = 0 POL(IFLOW(x_1, x_2, x_3)) = [1] + x_1*x_3 POL(LE(x_1, x_2)) = 0 POL(LOW(x_1, x_2)) = [2] + x_2 POL(QUICKSORT(x_1)) = [2]x_1 + x_1^2 POL(app(x_1, x_2)) = [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_3)) = x_1 + x_2 + x_3 POL(c15(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(false) = [1] POL(high(x_1, x_2)) = x_2 POL(ifhigh(x_1, x_2, x_3)) = x_3 POL(iflow(x_1, x_2, x_3)) = x_1*x_3 POL(le(x_1, x_2)) = [1] POL(low(x_1, x_2)) = x_2 POL(nil) = 0 POL(quicksort(x_1)) = 0 POL(s(x_1)) = 0 POL(true) = [1] ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, HIGH_2, IFHIGH_3, QUICKSORT_1, LOW_2 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c10_2, c11_1, c12_1, c14_3, c15_3, c6_2, c6_1 ---------------------------------------- (69) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, HIGH_2, IFHIGH_3, QUICKSORT_1, LOW_2 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c10_2, c11_1, c12_1, c14_3, c15_3, c6_2, c6_1 ---------------------------------------- (71) 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)) -> c2(LE(z0, z1)) We considered the (Usable) Rules: low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) le(0, z0) -> true high(z0, nil) -> nil ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) 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 + x_1*x_2 POL(IFHIGH(x_1, x_2, x_3)) = x_2*x_3 POL(IFLOW(x_1, x_2, x_3)) = 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_2 + x_1*x_2 POL(QUICKSORT(x_1)) = x_1^2 POL(app(x_1, x_2)) = [2] 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_3)) = x_1 + x_2 + x_3 POL(c15(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [2] + x_1 + x_2 POL(false) = [1] POL(high(x_1, x_2)) = x_2 POL(ifhigh(x_1, x_2, x_3)) = x_3 POL(iflow(x_1, x_2, x_3)) = x_1*x_3 POL(le(x_1, x_2)) = [1] POL(low(x_1, x_2)) = x_2 POL(nil) = 0 POL(quicksort(x_1)) = 0 POL(s(x_1)) = [2] + x_1 POL(true) = [1] ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, HIGH_2, IFHIGH_3, QUICKSORT_1, LOW_2 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c10_2, c11_1, c12_1, c14_3, c15_3, c6_2, c6_1 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) by HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2)), LE(0, z0)) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2)), LE(s(z0), 0)) HIGH(s(z1), cons(s(z0), x2)) -> c10(IFHIGH(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2)), LE(0, z0)) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2)), LE(s(z0), 0)) HIGH(s(z1), cons(s(z0), x2)) -> c10(IFHIGH(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, QUICKSORT_1, LOW_2, HIGH_2 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c14_3, c15_3, c6_2, c6_1, c10_2 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(s(z1), cons(s(z0), x2)) -> c10(IFHIGH(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, QUICKSORT_1, LOW_2, HIGH_2 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c14_3, c15_3, c6_2, c6_1, c10_2, c10_1 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) by QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil)), LOW(z0, nil)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil)), LOW(z0, nil)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(s(z1), cons(s(z0), x2)) -> c10(IFHIGH(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil)), LOW(z0, nil)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil)), LOW(z0, nil)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, QUICKSORT_1, LOW_2, HIGH_2 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c15_3, c6_2, c6_1, c10_2, c10_1, c14_3 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(s(z1), cons(s(z0), x2)) -> c10(IFHIGH(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, QUICKSORT_1, LOW_2, HIGH_2 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c15_3, c6_2, c6_1, c10_2, c10_1, c14_3, c14_2 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) by QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil)), HIGH(z0, nil)) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil)), HIGH(z0, nil)) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(s(z1), cons(s(z0), x2)) -> c10(IFHIGH(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil)), HIGH(z0, nil)) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil)), HIGH(z0, nil)) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_2, c6_1, c10_2, c10_1, c14_3, c14_2, c15_3 ---------------------------------------- (83) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(s(z1), cons(s(z0), x2)) -> c10(IFHIGH(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_2, c6_1, c10_2, c10_1, c14_3, c14_2, c15_3, c15_2 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) by LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(s(z1), cons(s(z0), x2)) -> c10(IFHIGH(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_2, c10_1, c14_3, c14_2, c15_3, c15_2, c6_2 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace HIGH(s(z1), cons(s(z0), x2)) -> c10(IFHIGH(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) by HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c14_3, c14_2, c15_3, c15_2, c6_2, c10_2 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) by QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c14_3, c14_2, c15_3, c15_2, c6_2, c10_2 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) by QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c14_2, c15_3, c15_2, c6_2, c10_2, c14_3 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) by QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c14_2, c15_3, c15_2, c6_2, c10_2, c14_3 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) by QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(x0, nil)) -> c14(APP(nil, cons(x0, quicksort(high(x0, nil)))), QUICKSORT(low(x0, nil))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(x0, nil)) -> c14(APP(nil, cons(x0, quicksort(high(x0, nil)))), QUICKSORT(low(x0, nil))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c15_3, c15_2, c6_2, c10_2, c14_3, c14_2 ---------------------------------------- (97) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(x0, nil)) -> c14(QUICKSORT(low(x0, nil))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c15_3, c15_2, c6_2, c10_2, c14_3, c14_2, c14_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(low(z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) by QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(x0, nil)) -> c14(QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c15_3, c15_2, c6_2, c10_2, c14_3, c14_2, c14_1 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(high(z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) by QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(x0, nil)) -> c14(QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c15_2, c6_2, c10_2, c14_3, c14_2, c14_1, c15_3 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(low(z0, nil)), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) by QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(x0, nil)) -> c14(QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c15_2, c6_2, c10_2, c14_3, c14_2, c14_1, c15_3 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) by QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(cons(x0, nil)) -> c15(APP(nil, cons(x0, quicksort(high(x0, nil)))), QUICKSORT(high(x0, nil))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(x0, nil)) -> c14(QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(cons(x0, nil)) -> c15(APP(nil, cons(x0, quicksort(high(x0, nil)))), QUICKSORT(high(x0, nil))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c14_1, c15_3, c15_2 ---------------------------------------- (107) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(x0, nil)) -> c14(QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(cons(x0, nil)) -> c15(QUICKSORT(high(x0, nil))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c14_1, c15_3, c15_2, c15_1 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(cons(x0, nil)) -> c14(QUICKSORT(low(x0, nil))) by QUICKSORT(cons(z0, nil)) -> c14(QUICKSORT(nil)) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(cons(x0, nil)) -> c15(QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(QUICKSORT(nil)) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: QUICKSORT(cons(z0, z1)) -> c14(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) QUICKSORT(cons(z0, z1)) -> c15(APP(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) HIGH(z0, cons(z1, z2)) -> c10(IFHIGH(le(z1, z0), z0, cons(z1, z2)), LE(z1, z0)) LOW(s(z1), cons(s(z0), x2)) -> c6(IFLOW(le(z0, z1), s(z1), cons(s(z0), x2)), LE(s(z0), s(z1))) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c15_3, c15_2, c15_1, c14_1 ---------------------------------------- (111) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: QUICKSORT(cons(z0, nil)) -> c14(QUICKSORT(nil)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(cons(x0, nil)) -> c15(QUICKSORT(high(x0, nil))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c15_3, c15_2, c15_1 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUICKSORT(cons(x0, nil)) -> c15(QUICKSORT(high(x0, nil))) by QUICKSORT(cons(z0, nil)) -> c15(QUICKSORT(nil)) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) QUICKSORT(cons(z0, nil)) -> c15(QUICKSORT(nil)) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c15_3, c15_2, c15_1 ---------------------------------------- (115) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: QUICKSORT(cons(z0, nil)) -> c15(QUICKSORT(nil)) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c15_3, c15_2 ---------------------------------------- (117) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IFLOW(true, z0, cons(z1, z2)) -> c7(LOW(z0, z2)) by IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFLOW_3, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1 Compound Symbols: c2_1, c4_1, c8_1, c11_1, c12_1, c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c15_3, c15_2, c7_1 ---------------------------------------- (119) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IFLOW(false, z0, cons(z1, z2)) -> c8(LOW(z0, z2)) by IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3 Compound Symbols: c2_1, c4_1, c11_1, c12_1, c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c15_3, c15_2, c7_1, c8_1 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) by QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3 Compound Symbols: c2_1, c4_1, c11_1, c12_1, c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c15_3, c15_2, c7_1, c8_1 ---------------------------------------- (123) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IFHIGH(true, z0, cons(z1, z2)) -> c11(HIGH(z0, z2)) by IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, IFHIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3 Compound Symbols: c2_1, c4_1, c12_1, c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c15_3, c15_2, c7_1, c8_1, c11_1 ---------------------------------------- (125) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IFHIGH(false, z0, cons(z1, z2)) -> c12(HIGH(z0, z2)) by IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3 Compound Symbols: c2_1, c4_1, c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c15_3, c15_2, c7_1, c8_1, c11_1, c12_1 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) by QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3 Compound Symbols: c2_1, c4_1, c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c15_3, c15_2, c7_1, c8_1, c11_1, c12_1 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) by QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) LE(s(z0), s(z1)) -> c2(LE(z0, z1)) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LE_2, APP_2, LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3 Compound Symbols: c2_1, c4_1, c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c15_3, c15_2, c7_1, c8_1, c11_1, c12_1 ---------------------------------------- (131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LE(s(z0), s(z1)) -> c2(LE(z0, z1)) by LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2)), LE(s(0), s(z0))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2)), LE(s(s(z0)), s(0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: APP_2, LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2 Compound Symbols: c4_1, c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c15_3, c15_2, c7_1, c8_1, c11_1, c12_1, c2_1 ---------------------------------------- (133) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) S tuples: APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: APP_2, LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2 Compound Symbols: c4_1, c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c15_3, c15_2, c7_1, c8_1, c11_1, c12_1, c2_1 ---------------------------------------- (135) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace APP(cons(z0, z1), z2) -> c4(APP(z1, z2)) by APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c15_3, c15_2, c7_1, c8_1, c11_1, c12_1, c2_1, c4_1 ---------------------------------------- (137) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) by QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c15_3, c15_2, c7_1, c8_1, c11_1, c12_1, c2_1, c4_1 ---------------------------------------- (139) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(low(z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) by QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c15_3, c15_2, c7_1, c8_1, c11_1, c12_1, c2_1, c4_1 ---------------------------------------- (141) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(z0, cons(0, x2))) -> c14(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(low(z0, cons(0, x2))), LOW(z0, cons(0, x2))) by QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c15_3, c15_2, c7_1, c8_1, c11_1, c12_1, c2_1, c4_1 ---------------------------------------- (143) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(0, cons(s(z0), x2))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(low(0, cons(s(z0), x2))), LOW(0, cons(s(z0), x2))) by QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c14_3, c14_2, c15_3, c15_2, c7_1, c8_1, c11_1, c12_1, c2_1, c4_1 ---------------------------------------- (145) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c14(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(low(s(z1), cons(s(z0), x2))), LOW(s(z1), cons(s(z0), x2))) by QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c14_2, c15_3, c15_2, c7_1, c8_1, c14_3, c11_1, c12_1, c2_1, c4_1 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(x0, nil)) -> c14(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(low(x0, nil))) by QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, nil)), QUICKSORT(nil)) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, nil)), QUICKSORT(nil)) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c14_2, c15_3, c15_2, c7_1, c8_1, c14_3, c11_1, c12_1, c2_1, c4_1 ---------------------------------------- (149) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, nil))) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c14_2, c15_3, c15_2, c7_1, c8_1, c14_3, c11_1, c12_1, c2_1, c4_1, c14_1 ---------------------------------------- (151) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) by QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(nil)) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(nil)) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c14_2, c15_3, c15_2, c7_1, c8_1, c14_3, c11_1, c12_1, c2_1, c4_1, c14_1 ---------------------------------------- (153) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil)))) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c14_2, c15_3, c15_2, c7_1, c8_1, c14_3, c11_1, c12_1, c2_1, c4_1, c14_1 ---------------------------------------- (155) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) by QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(nil)) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil)))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(nil)) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c15_3, c15_2, c7_1, c8_1, c14_3, c11_1, c12_1, c2_1, c4_1, c14_1, c14_2 ---------------------------------------- (157) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil)))) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c15_3, c15_2, c7_1, c8_1, c14_3, c11_1, c12_1, c2_1, c4_1, c14_1 ---------------------------------------- (159) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(low(z0, cons(0, x2))), cons(z0, quicksort(ifhigh(true, z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) by QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil)))) QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c15_3, c15_2, c7_1, c8_1, c14_3, c11_1, c12_1, c2_1, c4_1, c14_1 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(low(0, cons(s(z0), x2))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) by QUICKSORT(cons(0, cons(s(z0), z1))) -> c15(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(ifhigh(le(s(z0), 0), 0, cons(s(z0), z1))), HIGH(0, cons(s(z0), z1))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil)))) QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c15(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(ifhigh(le(s(z0), 0), 0, cons(s(z0), z1))), HIGH(0, cons(s(z0), z1))) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c15_3, c15_2, c7_1, c8_1, c14_3, c11_1, c12_1, c2_1, c4_1, c14_1 ---------------------------------------- (163) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(low(s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(ifhigh(le(z0, z1), s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) by QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c15(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(ifhigh(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), HIGH(s(z0), cons(s(z1), z2))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil)))) QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c15(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(ifhigh(le(s(z0), 0), 0, cons(s(z0), z1))), HIGH(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c15(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(ifhigh(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), HIGH(s(z0), cons(s(z1), z2))) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c15_3, c15_2, c7_1, c8_1, c14_3, c11_1, c12_1, c2_1, c4_1, c14_1 ---------------------------------------- (165) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) by QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(ifhigh(le(z1, z0), z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil)))) QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c15(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(ifhigh(le(s(z0), 0), 0, cons(s(z0), z1))), HIGH(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c15(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(ifhigh(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), HIGH(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(ifhigh(le(z1, z0), z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c15_3, c15_2, c7_1, c8_1, c14_3, c11_1, c12_1, c2_1, c4_1, c14_1 ---------------------------------------- (167) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(high(z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) by QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(ifhigh(le(z1, z0), z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil)))) QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c15(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(ifhigh(le(s(z0), 0), 0, cons(s(z0), z1))), HIGH(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c15(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(ifhigh(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), HIGH(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(ifhigh(le(z1, z0), z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c15_3, c15_2, c7_1, c8_1, c14_3, c11_1, c12_1, c2_1, c4_1, c14_1 ---------------------------------------- (169) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(z0, cons(0, x2))) -> c15(APP(quicksort(iflow(true, z0, cons(0, x2))), cons(z0, quicksort(high(z0, cons(0, x2))))), QUICKSORT(high(z0, cons(0, x2))), HIGH(z0, cons(0, x2))) by QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil)))) QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c15(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(ifhigh(le(s(z0), 0), 0, cons(s(z0), z1))), HIGH(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c15(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(ifhigh(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), HIGH(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(ifhigh(le(z1, z0), z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c15_3, c15_2, c7_1, c8_1, c14_3, c11_1, c12_1, c2_1, c4_1, c14_1 ---------------------------------------- (171) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(0, cons(s(z0), x2))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), x2))), cons(0, quicksort(high(0, cons(s(z0), x2))))), QUICKSORT(high(0, cons(s(z0), x2))), HIGH(0, cons(s(z0), x2))) by QUICKSORT(cons(0, cons(s(z0), z1))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(ifhigh(le(s(z0), 0), 0, cons(s(z0), z1))), HIGH(0, cons(s(z0), z1))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil)))) QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c15(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(ifhigh(le(s(z0), 0), 0, cons(s(z0), z1))), HIGH(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c15(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(ifhigh(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), HIGH(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(ifhigh(le(z1, z0), z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(ifhigh(le(s(z0), 0), 0, cons(s(z0), z1))), HIGH(0, cons(s(z0), z1))) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c15_3, c15_2, c7_1, c8_1, c14_3, c11_1, c12_1, c2_1, c4_1, c14_1 ---------------------------------------- (173) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LOW(s(x0), cons(s(x1), x2)) -> c6(LE(s(x1), s(x0))) by LOW(s(s(y1)), cons(s(s(y0)), z2)) -> c6(LE(s(s(y0)), s(s(y1)))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil)))) QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c15(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(ifhigh(le(s(z0), 0), 0, cons(s(z0), z1))), HIGH(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c15(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(ifhigh(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), HIGH(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(ifhigh(le(z1, z0), z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(ifhigh(le(s(z0), 0), 0, cons(s(z0), z1))), HIGH(0, cons(s(z0), z1))) LOW(s(s(y1)), cons(s(s(y0)), z2)) -> c6(LE(s(s(y0)), s(s(y1)))) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c15_3, c15_2, c7_1, c8_1, c14_3, c11_1, c12_1, c2_1, c4_1, c14_1 ---------------------------------------- (175) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(s(z1), cons(s(z0), x2))) -> c15(APP(quicksort(iflow(le(z0, z1), s(z1), cons(s(z0), x2))), cons(s(z1), quicksort(high(s(z1), cons(s(z0), x2))))), QUICKSORT(high(s(z1), cons(s(z0), x2))), HIGH(s(z1), cons(s(z0), x2))) by QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c15(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(ifhigh(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), HIGH(s(z0), cons(s(z1), z2))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil)))) QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c15(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(ifhigh(le(s(z0), 0), 0, cons(s(z0), z1))), HIGH(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c15(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(ifhigh(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), HIGH(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(ifhigh(le(z1, z0), z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(ifhigh(le(s(z0), 0), 0, cons(s(z0), z1))), HIGH(0, cons(s(z0), z1))) LOW(s(s(y1)), cons(s(s(y0)), z2)) -> c6(LE(s(s(y0)), s(s(y1)))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c15(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(ifhigh(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), HIGH(s(z0), cons(s(z1), z2))) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c15_2, c7_1, c8_1, c14_3, c11_1, c12_1, c2_1, c4_1, c14_1, c15_3 ---------------------------------------- (177) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUICKSORT(cons(x0, nil)) -> c15(APP(quicksort(low(x0, nil)), cons(x0, nil)), QUICKSORT(high(x0, nil))) by QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(low(z0, nil)), cons(z0, nil)), QUICKSORT(nil)) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil)))) QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c15(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(ifhigh(le(s(z0), 0), 0, cons(s(z0), z1))), HIGH(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c15(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(ifhigh(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), HIGH(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(ifhigh(le(z1, z0), z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(ifhigh(le(s(z0), 0), 0, cons(s(z0), z1))), HIGH(0, cons(s(z0), z1))) LOW(s(s(y1)), cons(s(s(y0)), z2)) -> c6(LE(s(s(y0)), s(s(y1)))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c15(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(ifhigh(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), HIGH(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(low(z0, nil)), cons(z0, nil)), QUICKSORT(nil)) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c15_2, c7_1, c8_1, c14_3, c11_1, c12_1, c2_1, c4_1, c14_1, c15_3 ---------------------------------------- (179) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) low(z0, nil) -> nil low(z0, cons(z1, z2)) -> iflow(le(z1, z0), z0, cons(z1, z2)) iflow(true, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) iflow(false, z0, cons(z1, z2)) -> low(z0, z2) high(z0, nil) -> nil high(z0, cons(z1, z2)) -> ifhigh(le(z1, z0), z0, cons(z1, z2)) ifhigh(true, z0, cons(z1, z2)) -> high(z0, z2) ifhigh(false, z0, cons(z1, z2)) -> cons(z1, high(z0, z2)) quicksort(nil) -> nil quicksort(cons(z0, z1)) -> app(quicksort(low(z0, z1)), cons(z0, quicksort(high(z0, z1)))) Tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) HIGH(z0, cons(0, x2)) -> c10(IFHIGH(true, z0, cons(0, x2))) HIGH(0, cons(s(z0), x2)) -> c10(IFHIGH(false, 0, cons(s(z0), x2))) LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c6(IFLOW(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c10(IFHIGH(le(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1)))) HIGH(s(x0), cons(s(x1), x2)) -> c10(LE(s(x1), s(x0))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(nil), cons(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOW(s(z0), cons(s(0), x2)) -> c6(IFLOW(true, s(z0), cons(s(0), x2))) LOW(s(0), cons(s(s(z0)), x2)) -> c6(IFLOW(false, s(0), cons(s(s(z0)), x2))) HIGH(s(z0), cons(s(0), x2)) -> c10(IFHIGH(true, s(z0), cons(s(0), x2))) HIGH(s(0), cons(s(s(z0)), x2)) -> c10(IFHIGH(false, s(0), cons(s(s(z0)), x2))) APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) QUICKSORT(cons(z0, cons(z1, z2))) -> c14(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(iflow(le(z1, z0), z0, cons(z1, z2))), LOW(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c14(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(iflow(le(0, z0), z0, cons(0, z1))), LOW(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c14(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(iflow(le(s(z0), 0), 0, cons(s(z0), z1))), LOW(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c14(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(iflow(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), LOW(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(low(z0, nil)), cons(z0, nil))) QUICKSORT(cons(z0, nil)) -> c14(APP(quicksort(nil), cons(z0, quicksort(nil)))) QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(low(z0, cons(0, z1))), cons(z0, quicksort(ifhigh(true, z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c15(APP(quicksort(low(0, cons(s(z0), z1))), cons(0, quicksort(ifhigh(false, 0, cons(s(z0), z1))))), QUICKSORT(ifhigh(le(s(z0), 0), 0, cons(s(z0), z1))), HIGH(0, cons(s(z0), z1))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c15(APP(quicksort(low(s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(ifhigh(le(z1, z0), s(z0), cons(s(z1), z2))))), QUICKSORT(ifhigh(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), HIGH(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, cons(z1, z2))) -> c15(APP(quicksort(iflow(le(z1, z0), z0, cons(z1, z2))), cons(z0, quicksort(ifhigh(le(z1, z0), z0, cons(z1, z2))))), QUICKSORT(ifhigh(le(z1, z0), z0, cons(z1, z2))), HIGH(z0, cons(z1, z2))) QUICKSORT(cons(z0, cons(0, z1))) -> c15(APP(quicksort(iflow(true, z0, cons(0, z1))), cons(z0, quicksort(high(z0, cons(0, z1))))), QUICKSORT(ifhigh(le(0, z0), z0, cons(0, z1))), HIGH(z0, cons(0, z1))) QUICKSORT(cons(0, cons(s(z0), z1))) -> c15(APP(quicksort(iflow(false, 0, cons(s(z0), z1))), cons(0, quicksort(high(0, cons(s(z0), z1))))), QUICKSORT(ifhigh(le(s(z0), 0), 0, cons(s(z0), z1))), HIGH(0, cons(s(z0), z1))) LOW(s(s(y1)), cons(s(s(y0)), z2)) -> c6(LE(s(s(y0)), s(s(y1)))) QUICKSORT(cons(s(z0), cons(s(z1), z2))) -> c15(APP(quicksort(iflow(le(z1, z0), s(z0), cons(s(z1), z2))), cons(s(z0), quicksort(high(s(z0), cons(s(z1), z2))))), QUICKSORT(ifhigh(le(s(z1), s(z0)), s(z0), cons(s(z1), z2))), HIGH(s(z0), cons(s(z1), z2))) QUICKSORT(cons(z0, nil)) -> c15(APP(quicksort(low(z0, nil)), cons(z0, nil))) S tuples: APP(cons(z0, cons(y0, y1)), z2) -> c4(APP(cons(y0, y1), z2)) K tuples: LOW(z0, cons(0, x2)) -> c6(IFLOW(true, z0, cons(0, x2))) LOW(0, cons(s(z0), x2)) -> c6(IFLOW(false, 0, cons(s(z0), x2))) IFLOW(true, x0, cons(0, x1)) -> c7(LOW(x0, x1)) IFLOW(true, s(x0), cons(s(0), x1)) -> c7(LOW(s(x0), x1)) IFLOW(true, s(s(x0)), cons(s(s(x1)), x2)) -> c7(LOW(s(s(x0)), x2)) IFLOW(false, 0, cons(s(x0), x1)) -> c8(LOW(0, x1)) IFLOW(false, s(0), cons(s(s(x0)), x1)) -> c8(LOW(s(0), x1)) IFLOW(false, s(s(x0)), cons(s(s(x1)), x2)) -> c8(LOW(s(s(x0)), x2)) IFHIGH(true, x0, cons(0, x1)) -> c11(HIGH(x0, x1)) IFHIGH(true, s(x0), cons(s(0), x1)) -> c11(HIGH(s(x0), x1)) IFHIGH(true, s(s(x0)), cons(s(s(x1)), x2)) -> c11(HIGH(s(s(x0)), x2)) IFHIGH(false, 0, cons(s(x0), x1)) -> c12(HIGH(0, x1)) IFHIGH(false, s(0), cons(s(s(x0)), x1)) -> c12(HIGH(s(0), x1)) IFHIGH(false, s(s(x0)), cons(s(s(x1)), x2)) -> c12(HIGH(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) Defined Rule Symbols: le_2, app_2, low_2, iflow_3, high_2, ifhigh_3, quicksort_1 Defined Pair Symbols: LOW_2, HIGH_2, QUICKSORT_1, IFLOW_3, IFHIGH_3, LE_2, APP_2 Compound Symbols: c6_1, c10_1, c6_2, c10_2, c15_2, c7_1, c8_1, c14_3, c11_1, c12_1, c2_1, c4_1, c14_1, c15_3, c15_1