KILLED proof of input_XiAmXpwWp6.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 18 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 17 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 13 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 402 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 101 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 325 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 157 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 164 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 116 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 1031 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 417 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 1043 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 410 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 6650 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 3253 ms] (54) CpxRNTS (55) CompletionProof [UPPER BOUND(ID), 0 ms] (56) CpxTypedWeightedCompleteTrs (57) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 18 ms] (58) CpxRNTS (59) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (60) CdtProblem (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 205 ms] (68) CdtProblem (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 64 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRhsSimplificationProcessorProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 94 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 106 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 106 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 3 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 103 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 132 ms] (120) CdtProblem (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 1 ms] (142) CdtProblem (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (146) CdtProblem (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) 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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (192) 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: qsort(nil) -> nil qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0) -> true ge(0, s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) last(nil) -> 0 last(cons(x, nil)) -> x last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 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: qsort(nil) -> nil qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) last(nil) -> 0' last(cons(x, nil)) -> x last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 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: qsort(nil) -> nil qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0) -> true ge(0, s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) last(nil) -> 0 last(cons(x, nil)) -> x last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 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: qsort(nil) -> nil [1] qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) [1] filterlow(n, nil) -> nil [1] filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) [1] if1(true, n, x, xs) -> filterlow(n, xs) [1] if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1] filterhigh(n, nil) -> nil [1] filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) [1] if2(true, n, x, xs) -> filterhigh(n, xs) [1] if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1] ge(x, 0) -> true [1] ge(0, s(x)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] append(nil, ys) -> ys [1] append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1] last(nil) -> 0 [1] last(cons(x, nil)) -> x [1] last(cons(x, cons(y, xs))) -> last(cons(y, xs)) [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: qsort(nil) -> nil [1] qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) [1] filterlow(n, nil) -> nil [1] filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) [1] if1(true, n, x, xs) -> filterlow(n, xs) [1] if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1] filterhigh(n, nil) -> nil [1] filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) [1] if2(true, n, x, xs) -> filterhigh(n, xs) [1] if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1] ge(x, 0) -> true [1] ge(0, s(x)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] append(nil, ys) -> ys [1] append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1] last(nil) -> 0 [1] last(cons(x, nil)) -> x [1] last(cons(x, cons(y, xs))) -> last(cons(y, xs)) [1] The TRS has the following type information: qsort :: nil:cons:ys -> nil:cons:ys nil :: nil:cons:ys cons :: 0:s -> nil:cons:ys -> nil:cons:ys append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys filterlow :: 0:s -> nil:cons:ys -> nil:cons:ys last :: nil:cons:ys -> 0:s filterhigh :: 0:s -> nil:cons:ys -> nil:cons:ys if1 :: true:false -> 0:s -> 0:s -> nil:cons:ys -> nil:cons:ys ge :: 0:s -> 0:s -> true:false true :: true:false false :: true:false if2 :: true:false -> 0:s -> 0:s -> nil:cons:ys -> nil:cons:ys 0 :: 0:s s :: 0:s -> 0:s ys :: nil:cons:ys 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: ge_2 qsort_1 filterlow_2 last_1 filterhigh_2 if1_4 if2_4 append_2 Due to the following rules being added: qsort(v0) -> null_qsort [0] filterlow(v0, v1) -> null_filterlow [0] last(v0) -> 0 [0] filterhigh(v0, v1) -> null_filterhigh [0] append(v0, v1) -> null_append [0] And the following fresh constants: null_qsort, null_filterlow, null_filterhigh, null_append ---------------------------------------- (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: qsort(nil) -> nil [1] qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) [1] filterlow(n, nil) -> nil [1] filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) [1] if1(true, n, x, xs) -> filterlow(n, xs) [1] if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1] filterhigh(n, nil) -> nil [1] filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) [1] if2(true, n, x, xs) -> filterhigh(n, xs) [1] if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1] ge(x, 0) -> true [1] ge(0, s(x)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] append(nil, ys) -> ys [1] append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1] last(nil) -> 0 [1] last(cons(x, nil)) -> x [1] last(cons(x, cons(y, xs))) -> last(cons(y, xs)) [1] qsort(v0) -> null_qsort [0] filterlow(v0, v1) -> null_filterlow [0] last(v0) -> 0 [0] filterhigh(v0, v1) -> null_filterhigh [0] append(v0, v1) -> null_append [0] The TRS has the following type information: qsort :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append nil :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append cons :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append append :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append filterlow :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append last :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> 0:s filterhigh :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append if1 :: true:false -> 0:s -> 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append ge :: 0:s -> 0:s -> true:false true :: true:false false :: true:false if2 :: true:false -> 0:s -> 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append 0 :: 0:s s :: 0:s -> 0:s ys :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append null_qsort :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append null_filterlow :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append null_filterhigh :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append null_append :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append 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: qsort(nil) -> nil [1] qsort(cons(x, nil)) -> append(qsort(filterlow(x, cons(x, nil))), cons(x, qsort(filterhigh(x, cons(x, nil))))) [4] qsort(cons(x, nil)) -> append(qsort(filterlow(x, cons(x, nil))), cons(x, qsort(filterhigh(0, cons(x, nil))))) [3] qsort(cons(x, nil)) -> append(qsort(filterlow(x, cons(x, nil))), cons(0, qsort(filterhigh(x, cons(x, nil))))) [3] qsort(cons(x, nil)) -> append(qsort(filterlow(x, cons(x, nil))), cons(0, qsort(filterhigh(0, cons(x, nil))))) [2] qsort(cons(x, cons(y', xs'))) -> append(qsort(filterlow(last(cons(y', xs')), cons(x, cons(y', xs')))), cons(last(cons(y', xs')), qsort(filterhigh(last(cons(y', xs')), cons(x, cons(y', xs')))))) [4] qsort(cons(x, cons(y', xs'))) -> append(qsort(filterlow(last(cons(y', xs')), cons(x, cons(y', xs')))), cons(last(cons(y', xs')), qsort(filterhigh(0, cons(x, cons(y', xs')))))) [3] qsort(cons(x, cons(y', xs'))) -> append(qsort(filterlow(last(cons(y', xs')), cons(x, cons(y', xs')))), cons(0, qsort(filterhigh(last(cons(y', xs')), cons(x, cons(y', xs')))))) [3] qsort(cons(x, cons(y', xs'))) -> append(qsort(filterlow(last(cons(y', xs')), cons(x, cons(y', xs')))), cons(0, qsort(filterhigh(0, cons(x, cons(y', xs')))))) [2] qsort(cons(x, nil)) -> append(qsort(filterlow(0, cons(x, nil))), cons(x, qsort(filterhigh(x, cons(x, nil))))) [3] qsort(cons(x, nil)) -> append(qsort(filterlow(0, cons(x, nil))), cons(x, qsort(filterhigh(0, cons(x, nil))))) [2] qsort(cons(x, cons(y'', xs''))) -> append(qsort(filterlow(0, cons(x, cons(y'', xs'')))), cons(last(cons(y'', xs'')), qsort(filterhigh(last(cons(y'', xs'')), cons(x, cons(y'', xs'')))))) [3] qsort(cons(x, cons(y'', xs''))) -> append(qsort(filterlow(0, cons(x, cons(y'', xs'')))), cons(last(cons(y'', xs'')), qsort(filterhigh(0, cons(x, cons(y'', xs'')))))) [2] qsort(cons(x, nil)) -> append(qsort(filterlow(0, cons(x, nil))), cons(0, qsort(filterhigh(x, cons(x, nil))))) [2] qsort(cons(x, cons(y1, xs1))) -> append(qsort(filterlow(0, cons(x, cons(y1, xs1)))), cons(0, qsort(filterhigh(last(cons(y1, xs1)), cons(x, cons(y1, xs1)))))) [2] qsort(cons(x, xs)) -> append(qsort(filterlow(0, cons(x, xs))), cons(0, qsort(filterhigh(0, cons(x, xs))))) [1] filterlow(n, nil) -> nil [1] filterlow(n, cons(0, xs)) -> if1(true, n, 0, xs) [2] filterlow(0, cons(s(x'), xs)) -> if1(false, 0, s(x'), xs) [2] filterlow(s(x''), cons(s(y2), xs)) -> if1(ge(x'', y2), s(x''), s(y2), xs) [2] if1(true, n, x, xs) -> filterlow(n, xs) [1] if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1] filterhigh(n, nil) -> nil [1] filterhigh(0, cons(x, xs)) -> if2(true, 0, x, xs) [2] filterhigh(s(x1), cons(0, xs)) -> if2(false, s(x1), 0, xs) [2] filterhigh(s(y3), cons(s(x2), xs)) -> if2(ge(x2, y3), s(y3), s(x2), xs) [2] if2(true, n, x, xs) -> filterhigh(n, xs) [1] if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1] ge(x, 0) -> true [1] ge(0, s(x)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] append(nil, ys) -> ys [1] append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1] last(nil) -> 0 [1] last(cons(x, nil)) -> x [1] last(cons(x, cons(y, xs))) -> last(cons(y, xs)) [1] qsort(v0) -> null_qsort [0] filterlow(v0, v1) -> null_filterlow [0] last(v0) -> 0 [0] filterhigh(v0, v1) -> null_filterhigh [0] append(v0, v1) -> null_append [0] The TRS has the following type information: qsort :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append nil :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append cons :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append append :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append filterlow :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append last :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> 0:s filterhigh :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append if1 :: true:false -> 0:s -> 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append ge :: 0:s -> 0:s -> true:false true :: true:false false :: true:false if2 :: true:false -> 0:s -> 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append 0 :: 0:s s :: 0:s -> 0:s ys :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append null_qsort :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append null_filterlow :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append null_filterhigh :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append null_append :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append 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: nil => 0 true => 1 false => 0 0 => 0 ys => 1 null_qsort => 0 null_filterlow => 0 null_filterhigh => 0 null_append => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 2 }-> if2(ge(x2, y3), 1 + y3, 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 1 + y3, y3 >= 0, x2 >= 0 filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 2 }-> if2(0, 1 + x1, 0, xs) :|: xs >= 0, x1 >= 0, z' = 1 + 0 + xs, z = 1 + x1 filterhigh(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 filterlow(z, z') -{ 2 }-> if1(ge(x'', y2), 1 + x'', 1 + y2, xs) :|: z = 1 + x'', xs >= 0, z' = 1 + (1 + y2) + xs, x'' >= 0, y2 >= 0 filterlow(z, z') -{ 2 }-> if1(1, n, 0, xs) :|: n >= 0, xs >= 0, z' = 1 + 0 + xs, z = n filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x ge(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> filterlow(n, xs) :|: n >= 0, xs >= 0, z = 1, x >= 0, z' = n, z'' = x, z1 = xs if1(z, z', z'', z1) -{ 1 }-> 1 + x + filterlow(n, xs) :|: n >= 0, xs >= 0, x >= 0, z' = n, z'' = x, z = 0, z1 = xs if2(z, z', z'', z1) -{ 1 }-> filterhigh(n, xs) :|: n >= 0, xs >= 0, z = 1, x >= 0, z' = n, z'' = x, z1 = xs if2(z, z', z'', z1) -{ 1 }-> 1 + x + filterhigh(n, xs) :|: n >= 0, xs >= 0, x >= 0, z' = n, z'' = x, z = 0, z1 = xs last(z) -{ 1 }-> x :|: x >= 0, z = 1 + x + 0 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 qsort(z) -{ 4 }-> append(qsort(filterlow(x, 1 + x + 0)), 1 + x + qsort(filterhigh(x, 1 + x + 0))) :|: x >= 0, z = 1 + x + 0 qsort(z) -{ 3 }-> append(qsort(filterlow(x, 1 + x + 0)), 1 + x + qsort(filterhigh(0, 1 + x + 0))) :|: x >= 0, z = 1 + x + 0 qsort(z) -{ 3 }-> append(qsort(filterlow(x, 1 + x + 0)), 1 + 0 + qsort(filterhigh(x, 1 + x + 0))) :|: x >= 0, z = 1 + x + 0 qsort(z) -{ 2 }-> append(qsort(filterlow(x, 1 + x + 0)), 1 + 0 + qsort(filterhigh(0, 1 + x + 0))) :|: x >= 0, z = 1 + x + 0 qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + 0)), 1 + x + qsort(filterhigh(x, 1 + x + 0))) :|: x >= 0, z = 1 + x + 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + 0)), 1 + x + qsort(filterhigh(0, 1 + x + 0))) :|: x >= 0, z = 1 + x + 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + 0)), 1 + 0 + qsort(filterhigh(x, 1 + x + 0))) :|: x >= 0, z = 1 + x + 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (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: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { append } { ge } { last } { filterlow, if1 } { filterhigh, if2 } { qsort } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {append}, {ge}, {last}, {filterlow,if1}, {filterhigh,if2}, {qsort} ---------------------------------------- (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: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {append}, {ge}, {last}, {filterlow,if1}, {filterhigh,if2}, {qsort} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: append after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {append}, {ge}, {last}, {filterlow,if1}, {filterhigh,if2}, {qsort} Previous analysis results are: append: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: append after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {ge}, {last}, {filterlow,if1}, {filterhigh,if2}, {qsort} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {ge}, {last}, {filterlow,if1}, {filterhigh,if2}, {qsort} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: ge after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {ge}, {last}, {filterlow,if1}, {filterhigh,if2}, {qsort} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: ?, size: O(1) [1] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: ge after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {last}, {filterlow,if1}, {filterhigh,if2}, {qsort} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 3 + z }-> if2(s'', 1 + (z - 1), 1 + x2, xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 4 + y2 }-> if1(s', 1 + (z - 1), 1 + y2, xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {last}, {filterlow,if1}, {filterhigh,if2}, {qsort} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: last 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: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 3 + z }-> if2(s'', 1 + (z - 1), 1 + x2, xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 4 + y2 }-> if1(s', 1 + (z - 1), 1 + y2, xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {last}, {filterlow,if1}, {filterhigh,if2}, {qsort} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] last: runtime: ?, size: O(n^1) [z] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: last after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 3 + z }-> if2(s'', 1 + (z - 1), 1 + x2, xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 4 + y2 }-> if1(s', 1 + (z - 1), 1 + y2, xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {filterlow,if1}, {filterhigh,if2}, {qsort} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] last: runtime: O(n^1) [2 + 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: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 3 + z }-> if2(s'', 1 + (z - 1), 1 + x2, xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 4 + y2 }-> if1(s', 1 + (z - 1), 1 + y2, xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 4 + xs + y }-> s14 :|: s14 >= 0, s14 <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 13 + 3*xs' + 3*y' }-> append(qsort(filterlow(s2, 1 + x + (1 + y' + xs'))), 1 + s3 + qsort(filterhigh(s4, 1 + x + (1 + y' + xs')))) :|: s2 >= 0, s2 <= 1 + y' + xs', s3 >= 0, s3 <= 1 + y' + xs', s4 >= 0, s4 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 9 + 2*xs' + 2*y' }-> append(qsort(filterlow(s5, 1 + x + (1 + y' + xs'))), 1 + s6 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s5 >= 0, s5 <= 1 + y' + xs', s6 >= 0, s6 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 9 + 2*xs' + 2*y' }-> append(qsort(filterlow(s7, 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(s8, 1 + x + (1 + y' + xs')))) :|: s7 >= 0, s7 <= 1 + y' + xs', s8 >= 0, s8 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 5 + xs' + y' }-> append(qsort(filterlow(s9, 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s9 >= 0, s9 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 9 + 2*xs'' + 2*y'' }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + s10 + qsort(filterhigh(s11, 1 + x + (1 + y'' + xs'')))) :|: s10 >= 0, s10 <= 1 + y'' + xs'', s11 >= 0, s11 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 5 + xs'' + y'' }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + s12 + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: s12 >= 0, s12 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 5 + xs1 + y1 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(s13, 1 + x + (1 + y1 + xs1)))) :|: s13 >= 0, s13 <= 1 + y1 + xs1, z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {filterlow,if1}, {filterhigh,if2}, {qsort} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] last: runtime: O(n^1) [2 + z], size: O(n^1) [z] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: filterlow after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' Computed SIZE bound using CoFloCo for: if1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z'' + z1 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 3 + z }-> if2(s'', 1 + (z - 1), 1 + x2, xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 4 + y2 }-> if1(s', 1 + (z - 1), 1 + y2, xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 4 + xs + y }-> s14 :|: s14 >= 0, s14 <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 13 + 3*xs' + 3*y' }-> append(qsort(filterlow(s2, 1 + x + (1 + y' + xs'))), 1 + s3 + qsort(filterhigh(s4, 1 + x + (1 + y' + xs')))) :|: s2 >= 0, s2 <= 1 + y' + xs', s3 >= 0, s3 <= 1 + y' + xs', s4 >= 0, s4 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 9 + 2*xs' + 2*y' }-> append(qsort(filterlow(s5, 1 + x + (1 + y' + xs'))), 1 + s6 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s5 >= 0, s5 <= 1 + y' + xs', s6 >= 0, s6 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 9 + 2*xs' + 2*y' }-> append(qsort(filterlow(s7, 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(s8, 1 + x + (1 + y' + xs')))) :|: s7 >= 0, s7 <= 1 + y' + xs', s8 >= 0, s8 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 5 + xs' + y' }-> append(qsort(filterlow(s9, 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s9 >= 0, s9 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 9 + 2*xs'' + 2*y'' }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + s10 + qsort(filterhigh(s11, 1 + x + (1 + y'' + xs'')))) :|: s10 >= 0, s10 <= 1 + y'' + xs'', s11 >= 0, s11 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 5 + xs'' + y'' }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + s12 + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: s12 >= 0, s12 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 5 + xs1 + y1 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(s13, 1 + x + (1 + y1 + xs1)))) :|: s13 >= 0, s13 <= 1 + y1 + xs1, z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {filterlow,if1}, {filterhigh,if2}, {qsort} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] last: runtime: O(n^1) [2 + z], size: O(n^1) [z] filterlow: runtime: ?, size: O(n^1) [z'] if1: runtime: ?, size: O(n^1) [1 + z'' + z1] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: filterlow after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 14 + 7*z' Computed RUNTIME bound using CoFloCo for: if1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 15 + 7*z1 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 3 + z }-> if2(s'', 1 + (z - 1), 1 + x2, xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 4 + y2 }-> if1(s', 1 + (z - 1), 1 + y2, xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 4 + xs + y }-> s14 :|: s14 >= 0, s14 <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 13 + 3*xs' + 3*y' }-> append(qsort(filterlow(s2, 1 + x + (1 + y' + xs'))), 1 + s3 + qsort(filterhigh(s4, 1 + x + (1 + y' + xs')))) :|: s2 >= 0, s2 <= 1 + y' + xs', s3 >= 0, s3 <= 1 + y' + xs', s4 >= 0, s4 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 9 + 2*xs' + 2*y' }-> append(qsort(filterlow(s5, 1 + x + (1 + y' + xs'))), 1 + s6 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s5 >= 0, s5 <= 1 + y' + xs', s6 >= 0, s6 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 9 + 2*xs' + 2*y' }-> append(qsort(filterlow(s7, 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(s8, 1 + x + (1 + y' + xs')))) :|: s7 >= 0, s7 <= 1 + y' + xs', s8 >= 0, s8 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 5 + xs' + y' }-> append(qsort(filterlow(s9, 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s9 >= 0, s9 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 9 + 2*xs'' + 2*y'' }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + s10 + qsort(filterhigh(s11, 1 + x + (1 + y'' + xs'')))) :|: s10 >= 0, s10 <= 1 + y'' + xs'', s11 >= 0, s11 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 5 + xs'' + y'' }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + s12 + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: s12 >= 0, s12 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 5 + xs1 + y1 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(s13, 1 + x + (1 + y1 + xs1)))) :|: s13 >= 0, s13 <= 1 + y1 + xs1, z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {filterhigh,if2}, {qsort} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] last: runtime: O(n^1) [2 + z], size: O(n^1) [z] filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z'] if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1] ---------------------------------------- (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: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 3 + z }-> if2(s'', 1 + (z - 1), 1 + x2, xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 10 + 7*z' }-> s30 :|: s30 >= 0, s30 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 17 + 7*xs }-> s31 :|: s31 >= 0, s31 <= 1 + x' + xs + 1, xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 19 + 7*xs + y2 }-> s32 :|: s32 >= 0, s32 <= 1 + y2 + xs + 1, s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s33 :|: s33 >= 0, s33 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s34 :|: s34 >= 0, s34 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 4 + xs + y }-> s14 :|: s14 >= 0, s14 <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 18 + 7*z }-> append(qsort(s15), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s15 >= 0, s15 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 17 + 7*z }-> append(qsort(s16), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: s16 >= 0, s16 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 17 + 7*z }-> append(qsort(s17), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s17 >= 0, s17 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 16 + 7*z }-> append(qsort(s18), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: s18 >= 0, s18 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 41 + 7*x + 10*xs' + 10*y' }-> append(qsort(s19), 1 + s3 + qsort(filterhigh(s4, 1 + x + (1 + y' + xs')))) :|: s19 >= 0, s19 <= 1 + x + (1 + y' + xs'), s2 >= 0, s2 <= 1 + y' + xs', s3 >= 0, s3 <= 1 + y' + xs', s4 >= 0, s4 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 37 + 7*x + 9*xs' + 9*y' }-> append(qsort(s20), 1 + s6 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s20 >= 0, s20 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= 1 + y' + xs', s6 >= 0, s6 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 37 + 7*x + 9*xs' + 9*y' }-> append(qsort(s21), 1 + 0 + qsort(filterhigh(s8, 1 + x + (1 + y' + xs')))) :|: s21 >= 0, s21 <= 1 + x + (1 + y' + xs'), s7 >= 0, s7 <= 1 + y' + xs', s8 >= 0, s8 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 33 + 7*x + 8*xs' + 8*y' }-> append(qsort(s22), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s22 >= 0, s22 <= 1 + x + (1 + y' + xs'), s9 >= 0, s9 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 17 + 7*z }-> append(qsort(s23), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s23 >= 0, s23 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 16 + 7*z }-> append(qsort(s24), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: s24 >= 0, s24 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 37 + 7*x + 9*xs'' + 9*y'' }-> append(qsort(s25), 1 + s10 + qsort(filterhigh(s11, 1 + x + (1 + y'' + xs'')))) :|: s25 >= 0, s25 <= 1 + x + (1 + y'' + xs''), s10 >= 0, s10 <= 1 + y'' + xs'', s11 >= 0, s11 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 33 + 7*x + 8*xs'' + 8*y'' }-> append(qsort(s26), 1 + s12 + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: s26 >= 0, s26 <= 1 + x + (1 + y'' + xs''), s12 >= 0, s12 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 16 + 7*z }-> append(qsort(s27), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s27 >= 0, s27 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 33 + 7*x + 8*xs1 + 8*y1 }-> append(qsort(s28), 1 + 0 + qsort(filterhigh(s13, 1 + x + (1 + y1 + xs1)))) :|: s28 >= 0, s28 <= 1 + x + (1 + y1 + xs1), s13 >= 0, s13 <= 1 + y1 + xs1, z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 22 + 7*x + 7*xs }-> append(qsort(s29), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: s29 >= 0, s29 <= 1 + x + xs, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {filterhigh,if2}, {qsort} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] last: runtime: O(n^1) [2 + z], size: O(n^1) [z] filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z'] if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: filterhigh after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' Computed SIZE bound using CoFloCo for: if2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z'' + z1 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 3 + z }-> if2(s'', 1 + (z - 1), 1 + x2, xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 10 + 7*z' }-> s30 :|: s30 >= 0, s30 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 17 + 7*xs }-> s31 :|: s31 >= 0, s31 <= 1 + x' + xs + 1, xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 19 + 7*xs + y2 }-> s32 :|: s32 >= 0, s32 <= 1 + y2 + xs + 1, s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s33 :|: s33 >= 0, s33 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s34 :|: s34 >= 0, s34 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 4 + xs + y }-> s14 :|: s14 >= 0, s14 <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 18 + 7*z }-> append(qsort(s15), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s15 >= 0, s15 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 17 + 7*z }-> append(qsort(s16), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: s16 >= 0, s16 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 17 + 7*z }-> append(qsort(s17), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s17 >= 0, s17 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 16 + 7*z }-> append(qsort(s18), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: s18 >= 0, s18 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 41 + 7*x + 10*xs' + 10*y' }-> append(qsort(s19), 1 + s3 + qsort(filterhigh(s4, 1 + x + (1 + y' + xs')))) :|: s19 >= 0, s19 <= 1 + x + (1 + y' + xs'), s2 >= 0, s2 <= 1 + y' + xs', s3 >= 0, s3 <= 1 + y' + xs', s4 >= 0, s4 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 37 + 7*x + 9*xs' + 9*y' }-> append(qsort(s20), 1 + s6 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s20 >= 0, s20 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= 1 + y' + xs', s6 >= 0, s6 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 37 + 7*x + 9*xs' + 9*y' }-> append(qsort(s21), 1 + 0 + qsort(filterhigh(s8, 1 + x + (1 + y' + xs')))) :|: s21 >= 0, s21 <= 1 + x + (1 + y' + xs'), s7 >= 0, s7 <= 1 + y' + xs', s8 >= 0, s8 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 33 + 7*x + 8*xs' + 8*y' }-> append(qsort(s22), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s22 >= 0, s22 <= 1 + x + (1 + y' + xs'), s9 >= 0, s9 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 17 + 7*z }-> append(qsort(s23), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s23 >= 0, s23 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 16 + 7*z }-> append(qsort(s24), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: s24 >= 0, s24 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 37 + 7*x + 9*xs'' + 9*y'' }-> append(qsort(s25), 1 + s10 + qsort(filterhigh(s11, 1 + x + (1 + y'' + xs'')))) :|: s25 >= 0, s25 <= 1 + x + (1 + y'' + xs''), s10 >= 0, s10 <= 1 + y'' + xs'', s11 >= 0, s11 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 33 + 7*x + 8*xs'' + 8*y'' }-> append(qsort(s26), 1 + s12 + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: s26 >= 0, s26 <= 1 + x + (1 + y'' + xs''), s12 >= 0, s12 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 16 + 7*z }-> append(qsort(s27), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s27 >= 0, s27 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 33 + 7*x + 8*xs1 + 8*y1 }-> append(qsort(s28), 1 + 0 + qsort(filterhigh(s13, 1 + x + (1 + y1 + xs1)))) :|: s28 >= 0, s28 <= 1 + x + (1 + y1 + xs1), s13 >= 0, s13 <= 1 + y1 + xs1, z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 22 + 7*x + 7*xs }-> append(qsort(s29), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: s29 >= 0, s29 <= 1 + x + xs, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {filterhigh,if2}, {qsort} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] last: runtime: O(n^1) [2 + z], size: O(n^1) [z] filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z'] if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1] filterhigh: runtime: ?, size: O(n^1) [z'] if2: runtime: ?, size: O(n^1) [1 + z'' + z1] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: filterhigh after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 11 + 3*z + z*z' + 4*z' Computed RUNTIME bound using KoAT for: if2 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 24 + 6*z' + 2*z'*z1 + 8*z1 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 3 + z }-> if2(s'', 1 + (z - 1), 1 + x2, xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 10 + 7*z' }-> s30 :|: s30 >= 0, s30 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 17 + 7*xs }-> s31 :|: s31 >= 0, s31 <= 1 + x' + xs + 1, xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 19 + 7*xs + y2 }-> s32 :|: s32 >= 0, s32 <= 1 + y2 + xs + 1, s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s33 :|: s33 >= 0, s33 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s34 :|: s34 >= 0, s34 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 4 + xs + y }-> s14 :|: s14 >= 0, s14 <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 18 + 7*z }-> append(qsort(s15), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s15 >= 0, s15 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 17 + 7*z }-> append(qsort(s16), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: s16 >= 0, s16 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 17 + 7*z }-> append(qsort(s17), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s17 >= 0, s17 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 16 + 7*z }-> append(qsort(s18), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: s18 >= 0, s18 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 41 + 7*x + 10*xs' + 10*y' }-> append(qsort(s19), 1 + s3 + qsort(filterhigh(s4, 1 + x + (1 + y' + xs')))) :|: s19 >= 0, s19 <= 1 + x + (1 + y' + xs'), s2 >= 0, s2 <= 1 + y' + xs', s3 >= 0, s3 <= 1 + y' + xs', s4 >= 0, s4 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 37 + 7*x + 9*xs' + 9*y' }-> append(qsort(s20), 1 + s6 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s20 >= 0, s20 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= 1 + y' + xs', s6 >= 0, s6 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 37 + 7*x + 9*xs' + 9*y' }-> append(qsort(s21), 1 + 0 + qsort(filterhigh(s8, 1 + x + (1 + y' + xs')))) :|: s21 >= 0, s21 <= 1 + x + (1 + y' + xs'), s7 >= 0, s7 <= 1 + y' + xs', s8 >= 0, s8 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 33 + 7*x + 8*xs' + 8*y' }-> append(qsort(s22), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s22 >= 0, s22 <= 1 + x + (1 + y' + xs'), s9 >= 0, s9 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 17 + 7*z }-> append(qsort(s23), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s23 >= 0, s23 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 16 + 7*z }-> append(qsort(s24), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: s24 >= 0, s24 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 37 + 7*x + 9*xs'' + 9*y'' }-> append(qsort(s25), 1 + s10 + qsort(filterhigh(s11, 1 + x + (1 + y'' + xs'')))) :|: s25 >= 0, s25 <= 1 + x + (1 + y'' + xs''), s10 >= 0, s10 <= 1 + y'' + xs'', s11 >= 0, s11 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 33 + 7*x + 8*xs'' + 8*y'' }-> append(qsort(s26), 1 + s12 + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: s26 >= 0, s26 <= 1 + x + (1 + y'' + xs''), s12 >= 0, s12 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 16 + 7*z }-> append(qsort(s27), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s27 >= 0, s27 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 33 + 7*x + 8*xs1 + 8*y1 }-> append(qsort(s28), 1 + 0 + qsort(filterhigh(s13, 1 + x + (1 + y1 + xs1)))) :|: s28 >= 0, s28 <= 1 + x + (1 + y1 + xs1), s13 >= 0, s13 <= 1 + y1 + xs1, z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 22 + 7*x + 7*xs }-> append(qsort(s29), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: s29 >= 0, s29 <= 1 + x + xs, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {qsort} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] last: runtime: O(n^1) [2 + z], size: O(n^1) [z] filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z'] if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1] filterhigh: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z'] if2: runtime: O(n^2) [24 + 6*z' + 2*z'*z1 + 8*z1], size: O(n^1) [1 + z'' + z1] ---------------------------------------- (49) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 26 + 8*xs }-> s50 :|: s50 >= 0, s50 <= x + xs + 1, xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 18 + 4*z + 2*z*z' + 8*z' }-> s51 :|: s51 >= 0, s51 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 27 + 8*xs + 2*xs*z + 7*z }-> s52 :|: s52 >= 0, s52 <= 1 + x2 + xs + 1, s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 10 + 7*z' }-> s30 :|: s30 >= 0, s30 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 17 + 7*xs }-> s31 :|: s31 >= 0, s31 <= 1 + x' + xs + 1, xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 19 + 7*xs + y2 }-> s32 :|: s32 >= 0, s32 <= 1 + y2 + xs + 1, s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s33 :|: s33 >= 0, s33 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s34 :|: s34 >= 0, s34 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> s53 :|: s53 >= 0, s53 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> 1 + z'' + s54 :|: s54 >= 0, s54 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 4 + xs + y }-> s14 :|: s14 >= 0, s14 <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 26 + 13*z + z^2 }-> append(qsort(s15), 1 + (z - 1) + qsort(s35)) :|: s35 >= 0, s35 <= 1 + (z - 1) + 0, s15 >= 0, s15 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 28 + 11*z }-> append(qsort(s16), 1 + (z - 1) + qsort(s36)) :|: s36 >= 0, s36 <= 1 + (z - 1) + 0, s16 >= 0, s16 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 25 + 13*z + z^2 }-> append(qsort(s17), 1 + 0 + qsort(s37)) :|: s37 >= 0, s37 <= 1 + (z - 1) + 0, s17 >= 0, s17 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 27 + 11*z }-> append(qsort(s18), 1 + 0 + qsort(s38)) :|: s38 >= 0, s38 <= 1 + (z - 1) + 0, s18 >= 0, s18 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 60 + 5*s4 + s4*x + s4*xs' + s4*y' + 11*x + 14*xs' + 14*y' }-> append(qsort(s19), 1 + s3 + qsort(s39)) :|: s39 >= 0, s39 <= 1 + x + (1 + y' + xs'), s19 >= 0, s19 <= 1 + x + (1 + y' + xs'), s2 >= 0, s2 <= 1 + y' + xs', s3 >= 0, s3 <= 1 + y' + xs', s4 >= 0, s4 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 56 + 11*x + 13*xs' + 13*y' }-> append(qsort(s20), 1 + s6 + qsort(s40)) :|: s40 >= 0, s40 <= 1 + x + (1 + y' + xs'), s20 >= 0, s20 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= 1 + y' + xs', s6 >= 0, s6 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 56 + 5*s8 + s8*x + s8*xs' + s8*y' + 11*x + 13*xs' + 13*y' }-> append(qsort(s21), 1 + 0 + qsort(s41)) :|: s41 >= 0, s41 <= 1 + x + (1 + y' + xs'), s21 >= 0, s21 <= 1 + x + (1 + y' + xs'), s7 >= 0, s7 <= 1 + y' + xs', s8 >= 0, s8 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 52 + 11*x + 12*xs' + 12*y' }-> append(qsort(s22), 1 + 0 + qsort(s42)) :|: s42 >= 0, s42 <= 1 + x + (1 + y' + xs'), s22 >= 0, s22 <= 1 + x + (1 + y' + xs'), s9 >= 0, s9 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 25 + 13*z + z^2 }-> append(qsort(s23), 1 + (z - 1) + qsort(s43)) :|: s43 >= 0, s43 <= 1 + (z - 1) + 0, s23 >= 0, s23 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 27 + 11*z }-> append(qsort(s24), 1 + (z - 1) + qsort(s44)) :|: s44 >= 0, s44 <= 1 + (z - 1) + 0, s24 >= 0, s24 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 56 + 5*s11 + s11*x + s11*xs'' + s11*y'' + 11*x + 13*xs'' + 13*y'' }-> append(qsort(s25), 1 + s10 + qsort(s45)) :|: s45 >= 0, s45 <= 1 + x + (1 + y'' + xs''), s25 >= 0, s25 <= 1 + x + (1 + y'' + xs''), s10 >= 0, s10 <= 1 + y'' + xs'', s11 >= 0, s11 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 52 + 11*x + 12*xs'' + 12*y'' }-> append(qsort(s26), 1 + s12 + qsort(s46)) :|: s46 >= 0, s46 <= 1 + x + (1 + y'' + xs''), s26 >= 0, s26 <= 1 + x + (1 + y'' + xs''), s12 >= 0, s12 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 24 + 13*z + z^2 }-> append(qsort(s27), 1 + 0 + qsort(s47)) :|: s47 >= 0, s47 <= 1 + (z - 1) + 0, s27 >= 0, s27 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 52 + 5*s13 + s13*x + s13*xs1 + s13*y1 + 11*x + 12*xs1 + 12*y1 }-> append(qsort(s28), 1 + 0 + qsort(s48)) :|: s48 >= 0, s48 <= 1 + x + (1 + y1 + xs1), s28 >= 0, s28 <= 1 + x + (1 + y1 + xs1), s13 >= 0, s13 <= 1 + y1 + xs1, z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 37 + 11*x + 11*xs }-> append(qsort(s29), 1 + 0 + qsort(s49)) :|: s49 >= 0, s49 <= 1 + x + xs, s29 >= 0, s29 <= 1 + x + xs, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {qsort} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] last: runtime: O(n^1) [2 + z], size: O(n^1) [z] filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z'] if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1] filterhigh: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z'] if2: runtime: O(n^2) [24 + 6*z' + 2*z'*z1 + 8*z1], size: O(n^1) [1 + z'' + z1] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: qsort after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 26 + 8*xs }-> s50 :|: s50 >= 0, s50 <= x + xs + 1, xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 18 + 4*z + 2*z*z' + 8*z' }-> s51 :|: s51 >= 0, s51 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 27 + 8*xs + 2*xs*z + 7*z }-> s52 :|: s52 >= 0, s52 <= 1 + x2 + xs + 1, s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 10 + 7*z' }-> s30 :|: s30 >= 0, s30 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 17 + 7*xs }-> s31 :|: s31 >= 0, s31 <= 1 + x' + xs + 1, xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 19 + 7*xs + y2 }-> s32 :|: s32 >= 0, s32 <= 1 + y2 + xs + 1, s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s33 :|: s33 >= 0, s33 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s34 :|: s34 >= 0, s34 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> s53 :|: s53 >= 0, s53 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> 1 + z'' + s54 :|: s54 >= 0, s54 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 4 + xs + y }-> s14 :|: s14 >= 0, s14 <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 26 + 13*z + z^2 }-> append(qsort(s15), 1 + (z - 1) + qsort(s35)) :|: s35 >= 0, s35 <= 1 + (z - 1) + 0, s15 >= 0, s15 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 28 + 11*z }-> append(qsort(s16), 1 + (z - 1) + qsort(s36)) :|: s36 >= 0, s36 <= 1 + (z - 1) + 0, s16 >= 0, s16 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 25 + 13*z + z^2 }-> append(qsort(s17), 1 + 0 + qsort(s37)) :|: s37 >= 0, s37 <= 1 + (z - 1) + 0, s17 >= 0, s17 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 27 + 11*z }-> append(qsort(s18), 1 + 0 + qsort(s38)) :|: s38 >= 0, s38 <= 1 + (z - 1) + 0, s18 >= 0, s18 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 60 + 5*s4 + s4*x + s4*xs' + s4*y' + 11*x + 14*xs' + 14*y' }-> append(qsort(s19), 1 + s3 + qsort(s39)) :|: s39 >= 0, s39 <= 1 + x + (1 + y' + xs'), s19 >= 0, s19 <= 1 + x + (1 + y' + xs'), s2 >= 0, s2 <= 1 + y' + xs', s3 >= 0, s3 <= 1 + y' + xs', s4 >= 0, s4 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 56 + 11*x + 13*xs' + 13*y' }-> append(qsort(s20), 1 + s6 + qsort(s40)) :|: s40 >= 0, s40 <= 1 + x + (1 + y' + xs'), s20 >= 0, s20 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= 1 + y' + xs', s6 >= 0, s6 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 56 + 5*s8 + s8*x + s8*xs' + s8*y' + 11*x + 13*xs' + 13*y' }-> append(qsort(s21), 1 + 0 + qsort(s41)) :|: s41 >= 0, s41 <= 1 + x + (1 + y' + xs'), s21 >= 0, s21 <= 1 + x + (1 + y' + xs'), s7 >= 0, s7 <= 1 + y' + xs', s8 >= 0, s8 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 52 + 11*x + 12*xs' + 12*y' }-> append(qsort(s22), 1 + 0 + qsort(s42)) :|: s42 >= 0, s42 <= 1 + x + (1 + y' + xs'), s22 >= 0, s22 <= 1 + x + (1 + y' + xs'), s9 >= 0, s9 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 25 + 13*z + z^2 }-> append(qsort(s23), 1 + (z - 1) + qsort(s43)) :|: s43 >= 0, s43 <= 1 + (z - 1) + 0, s23 >= 0, s23 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 27 + 11*z }-> append(qsort(s24), 1 + (z - 1) + qsort(s44)) :|: s44 >= 0, s44 <= 1 + (z - 1) + 0, s24 >= 0, s24 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 56 + 5*s11 + s11*x + s11*xs'' + s11*y'' + 11*x + 13*xs'' + 13*y'' }-> append(qsort(s25), 1 + s10 + qsort(s45)) :|: s45 >= 0, s45 <= 1 + x + (1 + y'' + xs''), s25 >= 0, s25 <= 1 + x + (1 + y'' + xs''), s10 >= 0, s10 <= 1 + y'' + xs'', s11 >= 0, s11 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 52 + 11*x + 12*xs'' + 12*y'' }-> append(qsort(s26), 1 + s12 + qsort(s46)) :|: s46 >= 0, s46 <= 1 + x + (1 + y'' + xs''), s26 >= 0, s26 <= 1 + x + (1 + y'' + xs''), s12 >= 0, s12 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 24 + 13*z + z^2 }-> append(qsort(s27), 1 + 0 + qsort(s47)) :|: s47 >= 0, s47 <= 1 + (z - 1) + 0, s27 >= 0, s27 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 52 + 5*s13 + s13*x + s13*xs1 + s13*y1 + 11*x + 12*xs1 + 12*y1 }-> append(qsort(s28), 1 + 0 + qsort(s48)) :|: s48 >= 0, s48 <= 1 + x + (1 + y1 + xs1), s28 >= 0, s28 <= 1 + x + (1 + y1 + xs1), s13 >= 0, s13 <= 1 + y1 + xs1, z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 37 + 11*x + 11*xs }-> append(qsort(s29), 1 + 0 + qsort(s49)) :|: s49 >= 0, s49 <= 1 + x + xs, s29 >= 0, s29 <= 1 + x + xs, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {qsort} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] last: runtime: O(n^1) [2 + z], size: O(n^1) [z] filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z'] if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1] filterhigh: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z'] if2: runtime: O(n^2) [24 + 6*z' + 2*z'*z1 + 8*z1], size: O(n^1) [1 + z'' + z1] qsort: runtime: ?, size: INF ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: qsort after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 26 + 8*xs }-> s50 :|: s50 >= 0, s50 <= x + xs + 1, xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 18 + 4*z + 2*z*z' + 8*z' }-> s51 :|: s51 >= 0, s51 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 27 + 8*xs + 2*xs*z + 7*z }-> s52 :|: s52 >= 0, s52 <= 1 + x2 + xs + 1, s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 10 + 7*z' }-> s30 :|: s30 >= 0, s30 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 17 + 7*xs }-> s31 :|: s31 >= 0, s31 <= 1 + x' + xs + 1, xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0 filterlow(z, z') -{ 19 + 7*xs + y2 }-> s32 :|: s32 >= 0, s32 <= 1 + y2 + xs + 1, s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0 filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s33 :|: s33 >= 0, s33 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s34 :|: s34 >= 0, s34 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> s53 :|: s53 >= 0, s53 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> 1 + z'' + s54 :|: s54 >= 0, s54 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0 last(z) -{ 4 + xs + y }-> s14 :|: s14 >= 0, s14 <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 qsort(z) -{ 26 + 13*z + z^2 }-> append(qsort(s15), 1 + (z - 1) + qsort(s35)) :|: s35 >= 0, s35 <= 1 + (z - 1) + 0, s15 >= 0, s15 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 28 + 11*z }-> append(qsort(s16), 1 + (z - 1) + qsort(s36)) :|: s36 >= 0, s36 <= 1 + (z - 1) + 0, s16 >= 0, s16 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 25 + 13*z + z^2 }-> append(qsort(s17), 1 + 0 + qsort(s37)) :|: s37 >= 0, s37 <= 1 + (z - 1) + 0, s17 >= 0, s17 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 27 + 11*z }-> append(qsort(s18), 1 + 0 + qsort(s38)) :|: s38 >= 0, s38 <= 1 + (z - 1) + 0, s18 >= 0, s18 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 60 + 5*s4 + s4*x + s4*xs' + s4*y' + 11*x + 14*xs' + 14*y' }-> append(qsort(s19), 1 + s3 + qsort(s39)) :|: s39 >= 0, s39 <= 1 + x + (1 + y' + xs'), s19 >= 0, s19 <= 1 + x + (1 + y' + xs'), s2 >= 0, s2 <= 1 + y' + xs', s3 >= 0, s3 <= 1 + y' + xs', s4 >= 0, s4 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 56 + 11*x + 13*xs' + 13*y' }-> append(qsort(s20), 1 + s6 + qsort(s40)) :|: s40 >= 0, s40 <= 1 + x + (1 + y' + xs'), s20 >= 0, s20 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= 1 + y' + xs', s6 >= 0, s6 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 56 + 5*s8 + s8*x + s8*xs' + s8*y' + 11*x + 13*xs' + 13*y' }-> append(qsort(s21), 1 + 0 + qsort(s41)) :|: s41 >= 0, s41 <= 1 + x + (1 + y' + xs'), s21 >= 0, s21 <= 1 + x + (1 + y' + xs'), s7 >= 0, s7 <= 1 + y' + xs', s8 >= 0, s8 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 52 + 11*x + 12*xs' + 12*y' }-> append(qsort(s22), 1 + 0 + qsort(s42)) :|: s42 >= 0, s42 <= 1 + x + (1 + y' + xs'), s22 >= 0, s22 <= 1 + x + (1 + y' + xs'), s9 >= 0, s9 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') qsort(z) -{ 25 + 13*z + z^2 }-> append(qsort(s23), 1 + (z - 1) + qsort(s43)) :|: s43 >= 0, s43 <= 1 + (z - 1) + 0, s23 >= 0, s23 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 27 + 11*z }-> append(qsort(s24), 1 + (z - 1) + qsort(s44)) :|: s44 >= 0, s44 <= 1 + (z - 1) + 0, s24 >= 0, s24 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 56 + 5*s11 + s11*x + s11*xs'' + s11*y'' + 11*x + 13*xs'' + 13*y'' }-> append(qsort(s25), 1 + s10 + qsort(s45)) :|: s45 >= 0, s45 <= 1 + x + (1 + y'' + xs''), s25 >= 0, s25 <= 1 + x + (1 + y'' + xs''), s10 >= 0, s10 <= 1 + y'' + xs'', s11 >= 0, s11 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 52 + 11*x + 12*xs'' + 12*y'' }-> append(qsort(s26), 1 + s12 + qsort(s46)) :|: s46 >= 0, s46 <= 1 + x + (1 + y'' + xs''), s26 >= 0, s26 <= 1 + x + (1 + y'' + xs''), s12 >= 0, s12 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0 qsort(z) -{ 24 + 13*z + z^2 }-> append(qsort(s27), 1 + 0 + qsort(s47)) :|: s47 >= 0, s47 <= 1 + (z - 1) + 0, s27 >= 0, s27 <= 1 + (z - 1) + 0, z - 1 >= 0 qsort(z) -{ 52 + 5*s13 + s13*x + s13*xs1 + s13*y1 + 11*x + 12*xs1 + 12*y1 }-> append(qsort(s28), 1 + 0 + qsort(s48)) :|: s48 >= 0, s48 <= 1 + x + (1 + y1 + xs1), s28 >= 0, s28 <= 1 + x + (1 + y1 + xs1), s13 >= 0, s13 <= 1 + y1 + xs1, z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0 qsort(z) -{ 37 + 11*x + 11*xs }-> append(qsort(s29), 1 + 0 + qsort(s49)) :|: s49 >= 0, s49 <= 1 + x + xs, s29 >= 0, s29 <= 1 + x + xs, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {qsort} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] last: runtime: O(n^1) [2 + z], size: O(n^1) [z] filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z'] if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1] filterhigh: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z'] if2: runtime: O(n^2) [24 + 6*z' + 2*z'*z1 + 8*z1], size: O(n^1) [1 + z'' + z1] qsort: runtime: INF, size: INF ---------------------------------------- (55) 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: qsort(v0) -> null_qsort [0] filterlow(v0, v1) -> null_filterlow [0] filterhigh(v0, v1) -> null_filterhigh [0] append(v0, v1) -> null_append [0] last(v0) -> null_last [0] ge(v0, v1) -> null_ge [0] if1(v0, v1, v2, v3) -> null_if1 [0] if2(v0, v1, v2, v3) -> null_if2 [0] And the following fresh constants: null_qsort, null_filterlow, null_filterhigh, null_append, null_last, null_ge, null_if1, null_if2 ---------------------------------------- (56) 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: qsort(nil) -> nil [1] qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) [1] filterlow(n, nil) -> nil [1] filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) [1] if1(true, n, x, xs) -> filterlow(n, xs) [1] if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1] filterhigh(n, nil) -> nil [1] filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) [1] if2(true, n, x, xs) -> filterhigh(n, xs) [1] if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1] ge(x, 0) -> true [1] ge(0, s(x)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] append(nil, ys) -> ys [1] append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1] last(nil) -> 0 [1] last(cons(x, nil)) -> x [1] last(cons(x, cons(y, xs))) -> last(cons(y, xs)) [1] qsort(v0) -> null_qsort [0] filterlow(v0, v1) -> null_filterlow [0] filterhigh(v0, v1) -> null_filterhigh [0] append(v0, v1) -> null_append [0] last(v0) -> null_last [0] ge(v0, v1) -> null_ge [0] if1(v0, v1, v2, v3) -> null_if1 [0] if2(v0, v1, v2, v3) -> null_if2 [0] The TRS has the following type information: qsort :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 nil :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 cons :: 0:s:null_last -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 append :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 filterlow :: 0:s:null_last -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 last :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> 0:s:null_last filterhigh :: 0:s:null_last -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 if1 :: true:false:null_ge -> 0:s:null_last -> 0:s:null_last -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 ge :: 0:s:null_last -> 0:s:null_last -> true:false:null_ge true :: true:false:null_ge false :: true:false:null_ge if2 :: true:false:null_ge -> 0:s:null_last -> 0:s:null_last -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 0 :: 0:s:null_last s :: 0:s:null_last -> 0:s:null_last ys :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 null_qsort :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 null_filterlow :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 null_filterhigh :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 null_append :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 null_last :: 0:s:null_last null_ge :: true:false:null_ge null_if1 :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 null_if2 :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 Rewrite Strategy: INNERMOST ---------------------------------------- (57) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 0 true => 2 false => 1 0 => 0 ys => 1 null_qsort => 0 null_filterlow => 0 null_filterhigh => 0 null_append => 0 null_last => 0 null_ge => 0 null_if1 => 0 null_if2 => 0 ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 1 }-> if2(ge(x, n), n, x, xs) :|: n >= 0, xs >= 0, z = n, z' = 1 + x + xs, x >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 filterlow(z, z') -{ 1 }-> if1(ge(n, x), n, x, xs) :|: n >= 0, xs >= 0, z = n, z' = 1 + x + xs, x >= 0 filterlow(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x ge(z, z') -{ 1 }-> 2 :|: x >= 0, z = x, z' = 0 ge(z, z') -{ 1 }-> 1 :|: z' = 1 + x, x >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 if1(z, z', z'', z1) -{ 1 }-> filterlow(n, xs) :|: z = 2, n >= 0, xs >= 0, x >= 0, z' = n, z'' = x, z1 = xs if1(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 if1(z, z', z'', z1) -{ 1 }-> 1 + x + filterlow(n, xs) :|: n >= 0, xs >= 0, z = 1, x >= 0, z' = n, z'' = x, z1 = xs if2(z, z', z'', z1) -{ 1 }-> filterhigh(n, xs) :|: z = 2, n >= 0, xs >= 0, x >= 0, z' = n, z'' = x, z1 = xs if2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + x + filterhigh(n, xs) :|: n >= 0, xs >= 0, z = 1, x >= 0, z' = n, z'' = x, z1 = xs last(z) -{ 1 }-> x :|: x >= 0, z = 1 + x + 0 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 qsort(z) -{ 1 }-> append(qsort(filterlow(last(1 + x + xs), 1 + x + xs)), 1 + last(1 + x + xs) + qsort(filterhigh(last(1 + x + xs), 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> 0 :|: z = 0 qsort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (59) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: qsort(nil) -> nil qsort(cons(z0, z1)) -> append(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))) filterlow(z0, nil) -> nil filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, nil) -> nil filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) append(nil, ys) -> ys append(cons(z0, z1), ys) -> cons(z0, append(z1, ys)) last(nil) -> 0 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: QSORT(nil) -> c QSORT(cons(z0, z1)) -> c1(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) FILTERLOW(z0, nil) -> c4 FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) FILTERHIGH(z0, nil) -> c8 FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(z0, 0) -> c12 GE(0, s(z0)) -> c13 GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(nil, ys) -> c15 APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(nil) -> c17 LAST(cons(z0, nil)) -> c18 LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) S tuples: QSORT(nil) -> c QSORT(cons(z0, z1)) -> c1(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) FILTERLOW(z0, nil) -> c4 FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) FILTERHIGH(z0, nil) -> c8 FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(z0, 0) -> c12 GE(0, s(z0)) -> c13 GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(nil, ys) -> c15 APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(nil) -> c17 LAST(cons(z0, nil)) -> c18 LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) K tuples:none Defined Rule Symbols: qsort_1, filterlow_2, if1_4, filterhigh_2, if2_4, ge_2, append_2, last_1 Defined Pair Symbols: QSORT_1, FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LAST_1 Compound Symbols: c, c1_4, c2_2, c3_4, c4, c5_2, c6_1, c7_1, c8, c9_2, c10_1, c11_1, c12, c13, c14_1, c15, c16_1, c17, c18, c19_1 ---------------------------------------- (61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 8 trailing nodes: GE(z0, 0) -> c12 FILTERHIGH(z0, nil) -> c8 FILTERLOW(z0, nil) -> c4 APPEND(nil, ys) -> c15 QSORT(nil) -> c LAST(cons(z0, nil)) -> c18 LAST(nil) -> c17 GE(0, s(z0)) -> c13 ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: qsort(nil) -> nil qsort(cons(z0, z1)) -> append(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))) filterlow(z0, nil) -> nil filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, nil) -> nil filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) append(nil, ys) -> ys append(cons(z0, z1), ys) -> cons(z0, append(z1, ys)) last(nil) -> 0 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: QSORT(cons(z0, z1)) -> c1(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) S tuples: QSORT(cons(z0, z1)) -> c1(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) K tuples:none Defined Rule Symbols: qsort_1, filterlow_2, if1_4, filterhigh_2, if2_4, ge_2, append_2, last_1 Defined Pair Symbols: QSORT_1, FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LAST_1 Compound Symbols: c1_4, c2_2, c3_4, c5_2, c6_1, c7_1, c9_2, c10_1, c11_1, c14_1, c16_1, c19_1 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: qsort(nil) -> nil qsort(cons(z0, z1)) -> append(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))) filterlow(z0, nil) -> nil filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, nil) -> nil filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) append(nil, ys) -> ys append(cons(z0, z1), ys) -> cons(z0, append(z1, ys)) last(nil) -> 0 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) S tuples: FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) K tuples:none Defined Rule Symbols: qsort_1, filterlow_2, if1_4, filterhigh_2, if2_4, ge_2, append_2, last_1 Defined Pair Symbols: FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1 Compound Symbols: c5_2, c6_1, c7_1, c9_2, c10_1, c11_1, c14_1, c16_1, c19_1, c1_3, c2_1, c3_3 ---------------------------------------- (65) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: qsort(nil) -> nil qsort(cons(z0, z1)) -> append(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))) append(nil, ys) -> ys append(cons(z0, z1), ys) -> cons(z0, append(z1, ys)) last(nil) -> 0 ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) S tuples: FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) K tuples:none Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1 Compound Symbols: c5_2, c6_1, c7_1, c9_2, c10_1, c11_1, c14_1, c16_1, c19_1, c1_3, c2_1, c3_3 ---------------------------------------- (67) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) We considered the (Usable) Rules:none And the Tuples: FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(APPEND(x_1, x_2)) = 0 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LAST(x_1)) = x_1 POL(QSORT(x_1)) = [1] POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = 0 POL(false) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_1 + x_2 POL(filterlow(x_1, x_2)) = [1] + x_1 POL(ge(x_1, x_2)) = x_1 + x_2 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4 POL(last(x_1)) = 0 POL(nil) = [1] POL(s(x_1)) = [1] + x_1 POL(true) = [1] POL(ys) = 0 ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) S tuples: FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1 Compound Symbols: c5_2, c6_1, c7_1, c9_2, c10_1, c11_1, c14_1, c16_1, c19_1, c1_3, c2_1, c3_3 ---------------------------------------- (69) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) We considered the (Usable) Rules:none And the Tuples: FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(APPEND(x_1, x_2)) = x_1 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LAST(x_1)) = 0 POL(QSORT(x_1)) = 0 POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = [1] + x_2 POL(false) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_1 + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = x_1 + x_2 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4 POL(last(x_1)) = 0 POL(nil) = [1] POL(s(x_1)) = [1] + x_1 POL(true) = [1] POL(ys) = 0 ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) S tuples: FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1 Compound Symbols: c5_2, c6_1, c7_1, c9_2, c10_1, c11_1, c14_1, c16_1, c19_1, c1_3, c2_1, c3_3 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) by FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2), GE(z0, 0)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2), GE(0, s(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2), GE(z0, 0)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2), GE(0, s(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2), GE(z0, 0)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2), GE(0, s(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2 Compound Symbols: c6_1, c7_1, c9_2, c10_1, c11_1, c14_1, c16_1, c19_1, c1_3, c2_1, c3_3, c5_2 ---------------------------------------- (73) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2 Compound Symbols: c6_1, c7_1, c9_2, c10_1, c11_1, c14_1, c16_1, c19_1, c1_3, c2_1, c3_3, c5_2, c5_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) by FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2), GE(z0, 0)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2), GE(0, s(z0))) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2), GE(z0, 0)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2), GE(0, s(z0))) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2), GE(z0, 0)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2), GE(0, s(z0))) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c1_3, c2_1, c3_3, c5_2, c5_1, c9_2 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c1_3, c2_1, c3_3, c5_2, c5_1, c9_2, c9_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) by QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c3_3, c5_2, c5_1, c9_2, c9_1, c1_3 ---------------------------------------- (81) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c3_3, c5_2, c5_1, c9_2, c9_1, c1_3, c1_2 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) by QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_2, c5_1, c9_2, c9_1, c1_3, c1_2, c3_3 ---------------------------------------- (85) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_2, c5_1, c9_2, c9_1, c1_3, c1_2, c3_3, c3_2 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) by FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_2, c9_1, c1_3, c1_2, c3_3, c3_2, c5_2 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) by FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c1_3, c1_2, c3_3, c3_2, c5_2, c9_2 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) by QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c1_3, c1_2, c3_3, c3_2, c5_2, c9_2 ---------------------------------------- (93) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c1_3, c1_2, c3_3, c3_2, c5_2, c9_2 ---------------------------------------- (95) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) We considered the (Usable) Rules: ge(s(z0), s(z1)) -> ge(z0, z1) filterhigh(z0, nil) -> nil filterlow(z0, nil) -> nil ge(0, s(z0)) -> false filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) ge(z0, 0) -> true if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) if1(true, z0, z1, z2) -> filterlow(z0, z2) if2(true, z0, z1, z2) -> filterhigh(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) And the Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APPEND(x_1, x_2)) = x_1 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = x_3 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LAST(x_1)) = 0 POL(QSORT(x_1)) = [1] + x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = [1] + x_2 POL(false) = [1] POL(filterhigh(x_1, x_2)) = x_2 POL(filterlow(x_1, x_2)) = x_2 POL(ge(x_1, x_2)) = [1] POL(if1(x_1, x_2, x_3, x_4)) = x_1 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = x_1 + x_4 POL(last(x_1)) = 0 POL(nil) = [1] POL(s(x_1)) = 0 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c1_3, c1_2, c3_3, c3_2, c5_2, c9_2 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c1(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))), LAST(cons(x0, cons(x1, x2)))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c1(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))), LAST(cons(x0, cons(x1, x2)))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c1(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))), LAST(cons(x0, cons(x1, x2)))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c1_2, c3_3, c3_2, c5_2, c9_2, c1_3 ---------------------------------------- (99) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c1_2, c3_3, c3_2, c5_2, c9_2, c1_3, c_1 ---------------------------------------- (101) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) We considered the (Usable) Rules:none And the Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APPEND(x_1, x_2)) = 0 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = x_3 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LAST(x_1)) = x_1 POL(QSORT(x_1)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = 0 POL(false) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_1 + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4 POL(last(x_1)) = 0 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c1_2, c3_3, c3_2, c5_2, c9_2, c1_3, c_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) by QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c3_3, c3_2, c5_2, c9_2, c1_3, c1_2, c_1, c1_1 ---------------------------------------- (105) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) We considered the (Usable) Rules:none And the Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APPEND(x_1, x_2)) = 0 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = x_3 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LAST(x_1)) = x_1 POL(QSORT(x_1)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = 0 POL(false) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_1 + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4 POL(last(x_1)) = 0 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c3_3, c3_2, c5_2, c9_2, c1_3, c1_2, c_1, c1_1 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) by QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c3_3, c3_2, c5_2, c9_2, c1_3, c1_2, c_1, c1_1 ---------------------------------------- (109) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c3_3, c3_2, c5_2, c9_2, c1_3, c1_2, c_1, c1_1 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c3(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))), LAST(cons(x0, cons(x1, x2)))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c3(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))), LAST(cons(x0, cons(x1, x2)))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c3(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))), LAST(cons(x0, cons(x1, x2)))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c3_2, c5_2, c9_2, c1_3, c1_2, c_1, c1_1, c3_3 ---------------------------------------- (113) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c3_2, c5_2, c9_2, c1_3, c1_2, c_1, c1_1, c3_3, c4_1 ---------------------------------------- (115) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) We considered the (Usable) Rules:none And the Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APPEND(x_1, x_2)) = 0 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = x_3 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LAST(x_1)) = x_1 POL(QSORT(x_1)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = 0 POL(false) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_1 + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4 POL(last(x_1)) = 0 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c3_2, c5_2, c9_2, c1_3, c1_2, c_1, c1_1, c3_3, c4_1 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) by QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_3, c1_2, c_1, c1_1, c3_3, c3_2, c4_1, c3_1 ---------------------------------------- (119) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) We considered the (Usable) Rules:none And the Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APPEND(x_1, x_2)) = 0 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = x_3 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LAST(x_1)) = x_1 POL(QSORT(x_1)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = 0 POL(false) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_1 + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4 POL(last(x_1)) = 0 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_3, c1_2, c_1, c1_1, c3_3, c3_2, c4_1, c3_1 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_3, c1_2, c_1, c1_1, c3_3, c3_2, c4_1, c3_1 ---------------------------------------- (123) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) by QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_3, c1_2, c_1, c1_1, c3_3, c3_2, c4_1, c3_1 ---------------------------------------- (125) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) S tuples: IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_2, c1_3, c_1, c1_1, c3_3, c3_2, c4_1, c3_1 ---------------------------------------- (127) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) by IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) S tuples: IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_2, c1_3, c_1, c1_1, c3_3, c3_2, c4_1, c3_1, c6_1 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) by QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) S tuples: IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_2, c1_3, c_1, c1_1, c3_3, c3_2, c4_1, c3_1, c6_1 ---------------------------------------- (131) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) by QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) S tuples: IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_3, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c6_1 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) by QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) S tuples: IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_3, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c6_1 ---------------------------------------- (135) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) by IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) S tuples: IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4 Compound Symbols: c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_3, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c6_1, c7_1 ---------------------------------------- (137) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) by QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) S tuples: IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4 Compound Symbols: c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_3, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c6_1, c7_1 ---------------------------------------- (139) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) by IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) S tuples: IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4 Compound Symbols: c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_3, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c6_1, c7_1, c10_1 ---------------------------------------- (141) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) by QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) S tuples: IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4 Compound Symbols: c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c1_3, c6_1, c7_1, c10_1 ---------------------------------------- (143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) by IF2(false, 0, z1, cons(y0, y1)) -> c11(FILTERHIGH(0, cons(y0, y1))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, 0, z1, cons(y0, y1)) -> c11(FILTERHIGH(0, cons(y0, y1))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) S tuples: GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, 0, z1, cons(y0, y1)) -> c11(FILTERHIGH(0, cons(y0, y1))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4 Compound Symbols: c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c1_3, c6_1, c7_1, c10_1, c11_1 ---------------------------------------- (145) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: IF2(false, 0, z1, cons(y0, y1)) -> c11(FILTERHIGH(0, cons(y0, y1))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) S tuples: GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4 Compound Symbols: c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c1_3, c6_1, c7_1, c10_1, c11_1 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) by QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: GE(s(z0), s(z1)) -> c14(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) S tuples: GE(s(z0), s(z1)) -> c14(GE(z0, z1)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4 Compound Symbols: c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c1_3, c6_1, c7_1, c10_1, c11_1 ---------------------------------------- (149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace GE(s(z0), s(z1)) -> c14(GE(z0, z1)) by GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c1_3, c6_1, c7_1, c10_1, c11_1, c14_1 ---------------------------------------- (151) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c1_3, c6_1, c7_1, c10_1, c11_1, c14_1 ---------------------------------------- (153) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) by QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c1_1, c3_3, c3_2, c4_1, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1 ---------------------------------------- (155) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) by QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c3_3, c3_2, c4_1, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1 ---------------------------------------- (157) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c3_3, c3_2, c4_1, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1 ---------------------------------------- (159) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c3_2, c3_3, c4_1, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) by QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c3_2, c3_3, c4_1, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1 ---------------------------------------- (163) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) by QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c3_3, c4_1, c3_2, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1 ---------------------------------------- (165) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) by QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c3_3, c4_1, c3_2, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1 ---------------------------------------- (167) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) by QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c3_3, c4_1, c3_2, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1 ---------------------------------------- (169) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) by QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c3_2, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1, c3_3 ---------------------------------------- (171) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) by QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c3_2, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1, c3_3 ---------------------------------------- (173) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) by QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1, c3_3, c3_2 ---------------------------------------- (175) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) by QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1, c3_3, c3_2, c3_1 ---------------------------------------- (177) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) by IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2))) IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil))) IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2))) IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2))) IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1, c3_3, c3_2, c3_1 ---------------------------------------- (179) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil))) IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2))) IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2))) IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1, c3_3, c3_2, c3_1 ---------------------------------------- (181) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil))) IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2))) IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2))) IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1, c3_3, c3_2, c3_1 ---------------------------------------- (183) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) by QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil))) IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2))) IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2))) IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1, c3_3, c3_2, c3_1 ---------------------------------------- (185) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) by QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil))) IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2))) IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2))) IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c1_3, c6_1, c7_1, c10_1, c11_1, c14_1, c1_2, c1_1, c3_3, c3_2, c3_1 ---------------------------------------- (187) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) by IF2(false, s(0), 0, cons(s(z1), z2)) -> c11(FILTERHIGH(s(0), cons(s(z1), z2))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil))) IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2))) IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(0), 0, cons(s(z1), z2)) -> c11(FILTERHIGH(s(0), cons(s(z1), z2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2))) IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(0), 0, cons(s(z1), z2)) -> c11(FILTERHIGH(s(0), cons(s(z1), z2))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c1_3, c6_1, c7_1, c10_1, c11_1, c14_1, c1_2, c1_1, c3_3, c3_2, c3_1 ---------------------------------------- (189) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) by IF2(false, s(s(z0)), 0, cons(s(0), z2)) -> c11(FILTERHIGH(s(s(z0)), cons(s(0), z2))) IF2(false, s(s(x0)), s(s(x1)), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2))) IF2(false, s(s(x0)), s(0), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil))) IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2))) IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(0), 0, cons(s(z1), z2)) -> c11(FILTERHIGH(s(0), cons(s(z1), z2))) IF2(false, s(s(z0)), 0, cons(s(0), z2)) -> c11(FILTERHIGH(s(s(z0)), cons(s(0), z2))) IF2(false, s(s(x0)), s(s(x1)), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2))) IF2(false, s(s(x0)), s(0), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2))) IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(0), 0, cons(s(z1), z2)) -> c11(FILTERHIGH(s(0), cons(s(z1), z2))) IF2(false, s(s(z0)), 0, cons(s(0), z2)) -> c11(FILTERHIGH(s(s(z0)), cons(s(0), z2))) IF2(false, s(s(x0)), s(s(x1)), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2))) IF2(false, s(s(x0)), s(0), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c1_3, c6_1, c7_1, c10_1, c11_1, c14_1, c1_2, c1_1, c3_3, c3_2, c3_1 ---------------------------------------- (191) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) by IF2(false, s(s(z0)), 0, cons(s(s(z2)), z3)) -> c11(FILTERHIGH(s(s(z0)), cons(s(s(z2)), z3))) IF2(false, s(s(x0)), s(s(x1)), cons(s(s(z2)), z3)) -> c11(FILTERHIGH(s(s(x0)), cons(s(s(z2)), z3))) IF2(false, s(s(x0)), s(0), cons(s(s(z2)), z3)) -> c11(FILTERHIGH(s(s(x0)), cons(s(s(z2)), z3))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2) filterhigh(z0, nil) -> nil if2(true, z0, z1, z2) -> filterhigh(z0, z2) if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2)) Tuples: APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil))) IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2))) IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(0), 0, cons(s(z1), z2)) -> c11(FILTERHIGH(s(0), cons(s(z1), z2))) IF2(false, s(s(z0)), 0, cons(s(0), z2)) -> c11(FILTERHIGH(s(s(z0)), cons(s(0), z2))) IF2(false, s(s(x0)), s(s(x1)), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2))) IF2(false, s(s(x0)), s(0), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2))) IF2(false, s(s(z0)), 0, cons(s(s(z2)), z3)) -> c11(FILTERHIGH(s(s(z0)), cons(s(s(z2)), z3))) IF2(false, s(s(x0)), s(s(x1)), cons(s(s(z2)), z3)) -> c11(FILTERHIGH(s(s(x0)), cons(s(s(z2)), z3))) IF2(false, s(s(x0)), s(0), cons(s(s(z2)), z3)) -> c11(FILTERHIGH(s(s(x0)), cons(s(s(z2)), z3))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0))) IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2))) GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2)) QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil))) QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil))) IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2))) IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2))) IF2(false, s(0), 0, cons(s(z1), z2)) -> c11(FILTERHIGH(s(0), cons(s(z1), z2))) IF2(false, s(s(z0)), 0, cons(s(0), z2)) -> c11(FILTERHIGH(s(s(z0)), cons(s(0), z2))) IF2(false, s(s(x0)), s(s(x1)), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2))) IF2(false, s(s(x0)), s(0), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2))) IF2(false, s(s(z0)), 0, cons(s(s(z2)), z3)) -> c11(FILTERHIGH(s(s(z0)), cons(s(s(z2)), z3))) IF2(false, s(s(x0)), s(s(x1)), cons(s(s(z2)), z3)) -> c11(FILTERHIGH(s(s(x0)), cons(s(s(z2)), z3))) IF2(false, s(s(x0)), s(0), cons(s(s(z2)), z3)) -> c11(FILTERHIGH(s(s(x0)), cons(s(s(z2)), z3))) K tuples: QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1))) APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys)) QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c1_3, c6_1, c7_1, c10_1, c11_1, c14_1, c1_2, c1_1, c3_3, c3_2, c3_1