KILLED proof of input_BUIWobVp7S.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 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), 384 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 90 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 215 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 49 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 958 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 446 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 1037 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 376 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 1508 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 518 ms] (48) CpxRNTS (49) CompletionProof [UPPER BOUND(ID), 0 ms] (50) CpxTypedWeightedCompleteTrs (51) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (54) CdtProblem (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 156 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 26 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 32 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms] (94) CdtProblem (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 2 ms] (172) CdtProblem (173) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (174) 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(x, cons(x, xs))), cons(x, qsort(filterhigh(x, 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)) 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(x, cons(x, xs))), cons(x, qsort(filterhigh(x, 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)) 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(x, cons(x, xs))), cons(x, qsort(filterhigh(x, 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)) 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(x, cons(x, xs))), cons(x, qsort(filterhigh(x, 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] 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(x, cons(x, xs))), cons(x, qsort(filterhigh(x, 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] 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 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: qsort_1 filterlow_2 filterhigh_2 ge_2 append_2 if1_4 if2_4 Due to the following rules being added: qsort(v0) -> null_qsort [0] filterlow(v0, v1) -> null_filterlow [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(x, cons(x, xs))), cons(x, qsort(filterhigh(x, 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] qsort(v0) -> null_qsort [0] filterlow(v0, v1) -> null_filterlow [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 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, xs)) -> append(qsort(if1(ge(x, x), x, x, xs)), cons(x, qsort(if2(ge(x, x), x, x, xs)))) [3] qsort(cons(x, xs)) -> append(qsort(if1(ge(x, x), x, x, xs)), cons(x, qsort(null_filterhigh))) [2] qsort(cons(x, xs)) -> append(qsort(null_filterlow), cons(x, qsort(if2(ge(x, x), x, x, xs)))) [2] qsort(cons(x, xs)) -> append(qsort(null_filterlow), cons(x, qsort(null_filterhigh))) [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(y'), xs)) -> if1(ge(x'', y'), s(x''), s(y'), 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(y''), cons(s(x2), xs)) -> if2(ge(x2, y''), s(y''), 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] qsort(v0) -> null_qsort [0] filterlow(v0, v1) -> null_filterlow [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 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, y''), 1 + y'', 1 + x2, xs) :|: xs >= 0, z = 1 + y'', z' = 1 + (1 + x2) + xs, y'' >= 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'', y'), 1 + x'', 1 + y', xs) :|: z = 1 + x'', xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, x'' >= 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 qsort(z) -{ 3 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 2 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 2 }-> append(qsort(0), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 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, y'), 1 + (z - 1), 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 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 qsort(z) -{ 3 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 2 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 2 }-> append(qsort(0), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 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 } { 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, y'), 1 + (z - 1), 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 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 qsort(z) -{ 3 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 2 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 2 }-> append(qsort(0), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: 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: {append}, {ge}, {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, y'), 1 + (z - 1), 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 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 qsort(z) -{ 3 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 2 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 2 }-> append(qsort(0), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: 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: {append}, {ge}, {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, y'), 1 + (z - 1), 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 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 qsort(z) -{ 3 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 2 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 2 }-> append(qsort(0), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: 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: {append}, {ge}, {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, y'), 1 + (z - 1), 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 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 qsort(z) -{ 3 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 2 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 2 }-> append(qsort(0), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: 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: {ge}, {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, y'), 1 + (z - 1), 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 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 qsort(z) -{ 3 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 2 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 2 }-> append(qsort(0), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: 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: {ge}, {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, y'), 1 + (z - 1), 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 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 qsort(z) -{ 3 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 2 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 2 }-> append(qsort(0), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: 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: {ge}, {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, y'), 1 + (z - 1), 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 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 qsort(z) -{ 3 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 2 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 2 }-> append(qsort(0), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: 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: {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(s4, 1 + (z - 1), 1 + x2, xs) :|: s4 >= 0, s4 <= 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 + y' }-> if1(s3, 1 + (z - 1), 1 + y', xs) :|: s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 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' }-> s5 :|: s5 >= 0, s5 <= 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 qsort(z) -{ 7 + 2*x }-> append(qsort(if1(s', x, x, xs)), 1 + x + qsort(if2(s'', x, x, xs))) :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 4 + x }-> append(qsort(if1(s1, x, x, xs)), 1 + x + qsort(0)) :|: s1 >= 0, s1 <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 4 + x }-> append(qsort(0), 1 + x + qsort(if2(s2, x, x, xs))) :|: s2 >= 0, s2 <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: 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: {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: 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 ---------------------------------------- (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(s4, 1 + (z - 1), 1 + x2, xs) :|: s4 >= 0, s4 <= 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 + y' }-> if1(s3, 1 + (z - 1), 1 + y', xs) :|: s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 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' }-> s5 :|: s5 >= 0, s5 <= 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 qsort(z) -{ 7 + 2*x }-> append(qsort(if1(s', x, x, xs)), 1 + x + qsort(if2(s'', x, x, xs))) :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 4 + x }-> append(qsort(if1(s1, x, x, xs)), 1 + x + qsort(0)) :|: s1 >= 0, s1 <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 4 + x }-> append(qsort(0), 1 + x + qsort(if2(s2, x, x, xs))) :|: s2 >= 0, s2 <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: 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: {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] filterlow: runtime: ?, size: O(n^1) [z'] if1: runtime: ?, size: O(n^1) [1 + z'' + z1] ---------------------------------------- (35) 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 ---------------------------------------- (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(s4, 1 + (z - 1), 1 + x2, xs) :|: s4 >= 0, s4 <= 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 + y' }-> if1(s3, 1 + (z - 1), 1 + y', xs) :|: s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 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' }-> s5 :|: s5 >= 0, s5 <= 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 qsort(z) -{ 7 + 2*x }-> append(qsort(if1(s', x, x, xs)), 1 + x + qsort(if2(s'', x, x, xs))) :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 4 + x }-> append(qsort(if1(s1, x, x, xs)), 1 + x + qsort(0)) :|: s1 >= 0, s1 <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 4 + x }-> append(qsort(0), 1 + x + qsort(if2(s2, x, x, xs))) :|: s2 >= 0, s2 <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: 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] 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] ---------------------------------------- (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(s4, 1 + (z - 1), 1 + x2, xs) :|: s4 >= 0, s4 <= 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') -{ 19 + 7*xs + y' }-> s10 :|: s10 >= 0, s10 <= 1 + y' + xs + 1, s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 filterlow(z, z') -{ 10 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 17 + 7*xs }-> s9 :|: s9 >= 0, s9 <= 1 + x' + xs + 1, 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' }-> s5 :|: s5 >= 0, s5 <= 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 }-> s11 :|: s11 >= 0, s11 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s12 :|: s12 >= 0, s12 <= 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 qsort(z) -{ 22 + 2*x + 7*xs }-> append(qsort(s6), 1 + x + qsort(if2(s'', x, x, xs))) :|: s6 >= 0, s6 <= x + xs + 1, s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 19 + x + 7*xs }-> append(qsort(s7), 1 + x + qsort(0)) :|: s7 >= 0, s7 <= x + xs + 1, s1 >= 0, s1 <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 4 + x }-> append(qsort(0), 1 + x + qsort(if2(s2, x, x, xs))) :|: s2 >= 0, s2 <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: 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] 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] ---------------------------------------- (39) 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 ---------------------------------------- (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(s4, 1 + (z - 1), 1 + x2, xs) :|: s4 >= 0, s4 <= 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') -{ 19 + 7*xs + y' }-> s10 :|: s10 >= 0, s10 <= 1 + y' + xs + 1, s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 filterlow(z, z') -{ 10 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 17 + 7*xs }-> s9 :|: s9 >= 0, s9 <= 1 + x' + xs + 1, 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' }-> s5 :|: s5 >= 0, s5 <= 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 }-> s11 :|: s11 >= 0, s11 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s12 :|: s12 >= 0, s12 <= 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 qsort(z) -{ 22 + 2*x + 7*xs }-> append(qsort(s6), 1 + x + qsort(if2(s'', x, x, xs))) :|: s6 >= 0, s6 <= x + xs + 1, s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 19 + x + 7*xs }-> append(qsort(s7), 1 + x + qsort(0)) :|: s7 >= 0, s7 <= x + xs + 1, s1 >= 0, s1 <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 4 + x }-> append(qsort(0), 1 + x + qsort(if2(s2, x, x, xs))) :|: s2 >= 0, s2 <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: 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] 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] ---------------------------------------- (41) 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 ---------------------------------------- (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(s4, 1 + (z - 1), 1 + x2, xs) :|: s4 >= 0, s4 <= 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') -{ 19 + 7*xs + y' }-> s10 :|: s10 >= 0, s10 <= 1 + y' + xs + 1, s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 filterlow(z, z') -{ 10 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 17 + 7*xs }-> s9 :|: s9 >= 0, s9 <= 1 + x' + xs + 1, 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' }-> s5 :|: s5 >= 0, s5 <= 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 }-> s11 :|: s11 >= 0, s11 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s12 :|: s12 >= 0, s12 <= 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 qsort(z) -{ 22 + 2*x + 7*xs }-> append(qsort(s6), 1 + x + qsort(if2(s'', x, x, xs))) :|: s6 >= 0, s6 <= x + xs + 1, s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 19 + x + 7*xs }-> append(qsort(s7), 1 + x + qsort(0)) :|: s7 >= 0, s7 <= x + xs + 1, s1 >= 0, s1 <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 4 + x }-> append(qsort(0), 1 + x + qsort(if2(s2, x, x, xs))) :|: s2 >= 0, s2 <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: 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] 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] ---------------------------------------- (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') -{ 26 + 8*xs }-> s15 :|: s15 >= 0, s15 <= x + xs + 1, xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 18 + 4*z + 2*z*z' + 8*z' }-> s16 :|: s16 >= 0, s16 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 27 + 8*xs + 2*xs*z + 7*z }-> s17 :|: s17 >= 0, s17 <= 1 + x2 + xs + 1, s4 >= 0, s4 <= 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') -{ 19 + 7*xs + y' }-> s10 :|: s10 >= 0, s10 <= 1 + y' + xs + 1, s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 filterlow(z, z') -{ 10 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 17 + 7*xs }-> s9 :|: s9 >= 0, s9 <= 1 + x' + xs + 1, 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' }-> s5 :|: s5 >= 0, s5 <= 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 }-> s11 :|: s11 >= 0, s11 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s12 :|: s12 >= 0, s12 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> s18 :|: s18 >= 0, s18 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> 1 + z'' + s19 :|: s19 >= 0, s19 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0 qsort(z) -{ 46 + 8*x + 2*x*xs + 15*xs }-> append(qsort(s6), 1 + x + qsort(s13)) :|: s13 >= 0, s13 <= x + xs + 1, s6 >= 0, s6 <= x + xs + 1, s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 19 + x + 7*xs }-> append(qsort(s7), 1 + x + qsort(0)) :|: s7 >= 0, s7 <= x + xs + 1, s1 >= 0, s1 <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 28 + 7*x + 2*x*xs + 8*xs }-> append(qsort(0), 1 + x + qsort(s14)) :|: s14 >= 0, s14 <= x + xs + 1, s2 >= 0, s2 <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: 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] 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] ---------------------------------------- (45) 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: ? ---------------------------------------- (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') -{ 26 + 8*xs }-> s15 :|: s15 >= 0, s15 <= x + xs + 1, xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 18 + 4*z + 2*z*z' + 8*z' }-> s16 :|: s16 >= 0, s16 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 27 + 8*xs + 2*xs*z + 7*z }-> s17 :|: s17 >= 0, s17 <= 1 + x2 + xs + 1, s4 >= 0, s4 <= 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') -{ 19 + 7*xs + y' }-> s10 :|: s10 >= 0, s10 <= 1 + y' + xs + 1, s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 filterlow(z, z') -{ 10 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 17 + 7*xs }-> s9 :|: s9 >= 0, s9 <= 1 + x' + xs + 1, 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' }-> s5 :|: s5 >= 0, s5 <= 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 }-> s11 :|: s11 >= 0, s11 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s12 :|: s12 >= 0, s12 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> s18 :|: s18 >= 0, s18 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> 1 + z'' + s19 :|: s19 >= 0, s19 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0 qsort(z) -{ 46 + 8*x + 2*x*xs + 15*xs }-> append(qsort(s6), 1 + x + qsort(s13)) :|: s13 >= 0, s13 <= x + xs + 1, s6 >= 0, s6 <= x + xs + 1, s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 19 + x + 7*xs }-> append(qsort(s7), 1 + x + qsort(0)) :|: s7 >= 0, s7 <= x + xs + 1, s1 >= 0, s1 <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 28 + 7*x + 2*x*xs + 8*xs }-> append(qsort(0), 1 + x + qsort(s14)) :|: s14 >= 0, s14 <= x + xs + 1, s2 >= 0, s2 <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: 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] 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 ---------------------------------------- (47) 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: ? ---------------------------------------- (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') -{ 26 + 8*xs }-> s15 :|: s15 >= 0, s15 <= x + xs + 1, xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 filterhigh(z, z') -{ 18 + 4*z + 2*z*z' + 8*z' }-> s16 :|: s16 >= 0, s16 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 filterhigh(z, z') -{ 27 + 8*xs + 2*xs*z + 7*z }-> s17 :|: s17 >= 0, s17 <= 1 + x2 + xs + 1, s4 >= 0, s4 <= 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') -{ 19 + 7*xs + y' }-> s10 :|: s10 >= 0, s10 <= 1 + y' + xs + 1, s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 filterlow(z, z') -{ 10 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 17 + 7*xs }-> s9 :|: s9 >= 0, s9 <= 1 + x' + xs + 1, 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' }-> s5 :|: s5 >= 0, s5 <= 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 }-> s11 :|: s11 >= 0, s11 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s12 :|: s12 >= 0, s12 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> s18 :|: s18 >= 0, s18 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> 1 + z'' + s19 :|: s19 >= 0, s19 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0 qsort(z) -{ 46 + 8*x + 2*x*xs + 15*xs }-> append(qsort(s6), 1 + x + qsort(s13)) :|: s13 >= 0, s13 <= x + xs + 1, s6 >= 0, s6 <= x + xs + 1, s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 19 + x + 7*xs }-> append(qsort(s7), 1 + x + qsort(0)) :|: s7 >= 0, s7 <= x + xs + 1, s1 >= 0, s1 <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 28 + 7*x + 2*x*xs + 8*xs }-> append(qsort(0), 1 + x + qsort(s14)) :|: s14 >= 0, s14 <= x + xs + 1, s2 >= 0, s2 <= 1, z = 1 + x + xs, xs >= 0, x >= 0 qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: 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] 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 ---------------------------------------- (49) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: qsort(v0) -> null_qsort [0] filterlow(v0, v1) -> null_filterlow [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 ---------------------------------------- (50) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: qsort(nil) -> nil [1] qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, 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] qsort(v0) -> null_qsort [0] filterlow(v0, v1) -> null_filterlow [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 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 ---------------------------------------- (51) 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 ---------------------------------------- (52) 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 }-> 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 qsort(z) -{ 1 }-> append(qsort(filterlow(x, 1 + x + xs)), 1 + x + qsort(filterhigh(x, 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. ---------------------------------------- (53) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: qsort(nil) -> nil qsort(cons(z0, z1)) -> append(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, 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)) Tuples: QSORT(nil) -> c QSORT(cons(z0, z1)) -> c1(APPEND(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1))))), QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(APPEND(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1))))), QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) FILTERLOW(z0, nil) -> c3 FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) FILTERHIGH(z0, nil) -> c7 FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(z0, 0) -> c11 GE(0, s(z0)) -> c12 GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(nil, ys) -> c14 APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) S tuples: QSORT(nil) -> c QSORT(cons(z0, z1)) -> c1(APPEND(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1))))), QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(APPEND(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1))))), QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) FILTERLOW(z0, nil) -> c3 FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) FILTERHIGH(z0, nil) -> c7 FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(z0, 0) -> c11 GE(0, s(z0)) -> c12 GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(nil, ys) -> c14 APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) K tuples:none Defined Rule Symbols: qsort_1, filterlow_2, if1_4, filterhigh_2, if2_4, ge_2, append_2 Defined Pair Symbols: QSORT_1, FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2 Compound Symbols: c, c1_3, c2_3, c3, c4_2, c5_1, c6_1, c7, c8_2, c9_1, c10_1, c11, c12, c13_1, c14, c15_1 ---------------------------------------- (55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing nodes: GE(0, s(z0)) -> c12 FILTERHIGH(z0, nil) -> c7 FILTERLOW(z0, nil) -> c3 APPEND(nil, ys) -> c14 GE(z0, 0) -> c11 QSORT(nil) -> c ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: qsort(nil) -> nil qsort(cons(z0, z1)) -> append(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, 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)) Tuples: QSORT(cons(z0, z1)) -> c1(APPEND(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1))))), QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(APPEND(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1))))), QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) S tuples: QSORT(cons(z0, z1)) -> c1(APPEND(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1))))), QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(APPEND(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1))))), QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) K tuples:none Defined Rule Symbols: qsort_1, filterlow_2, if1_4, filterhigh_2, if2_4, ge_2, append_2 Defined Pair Symbols: QSORT_1, FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2 Compound Symbols: c1_3, c2_3, c4_2, c5_1, c6_1, c8_2, c9_1, c10_1, c13_1, c15_1 ---------------------------------------- (57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: qsort(nil) -> nil qsort(cons(z0, z1)) -> append(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, 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)) Tuples: FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) S tuples: FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) K tuples:none Defined Rule Symbols: qsort_1, filterlow_2, if1_4, filterhigh_2, if2_4, ge_2, append_2 Defined Pair Symbols: FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, QSORT_1 Compound Symbols: c4_2, c5_1, c6_1, c8_2, c9_1, c10_1, c13_1, c15_1, c1_2, c2_2 ---------------------------------------- (59) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: qsort(nil) -> nil qsort(cons(z0, z1)) -> append(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1))))) append(nil, ys) -> ys append(cons(z0, z1), ys) -> cons(z0, append(z1, ys)) ---------------------------------------- (60) 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 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)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) S tuples: FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) K tuples:none Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, QSORT_1 Compound Symbols: c4_2, c5_1, c6_1, c8_2, c9_1, c10_1, c13_1, c15_1, c1_2, c2_2 ---------------------------------------- (61) 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) -> c15(APPEND(z1, ys)) We considered the (Usable) Rules:none And the Tuples: FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, 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(QSORT(x_1)) = 0 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_1 + x_2 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(nil) = [1] POL(s(x_1)) = [1] + x_1 POL(true) = [1] POL(ys) = 0 ---------------------------------------- (62) 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 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)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) S tuples: FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, QSORT_1 Compound Symbols: c4_2, c5_1, c6_1, c8_2, c9_1, c10_1, c13_1, c15_1, c1_2, c2_2 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) by FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2), GE(z0, 0)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2), GE(0, s(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) ---------------------------------------- (64) 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 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2), GE(z0, 0)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2), GE(0, s(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) S tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2), GE(z0, 0)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2), GE(0, s(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, QSORT_1, FILTERLOW_2 Compound Symbols: c5_1, c6_1, c8_2, c9_1, c10_1, c13_1, c15_1, c1_2, c2_2, c4_2 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) S tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, QSORT_1, FILTERLOW_2 Compound Symbols: c5_1, c6_1, c8_2, c9_1, c10_1, c13_1, c15_1, c1_2, c2_2, c4_2, c4_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) by FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2), GE(z0, 0)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2), GE(0, s(z0))) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) ---------------------------------------- (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 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2), GE(z0, 0)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2), GE(0, s(z0))) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) S tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2), GE(z0, 0)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2), GE(0, s(z0))) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c1_2, c2_2, c4_2, c4_1, c8_2 ---------------------------------------- (69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) S tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c1_2, c2_2, c4_2, c4_1, c8_2, c8_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) by QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) ---------------------------------------- (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 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) S tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c2_2, c4_2, c4_1, c8_2, c8_1, c1_2, c1_1 ---------------------------------------- (73) 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, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) We considered the (Usable) Rules:none And the Tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) 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(QSORT(x_1)) = [1] POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = 0 POL(false) = 0 POL(filterhigh(x_1, x_2)) = x_1 POL(filterlow(x_1, x_2)) = [1] + x_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)) = x_2 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (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 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) S tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2))) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c2_2, c4_2, c4_1, c8_2, c8_1, c1_2, c1_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) by QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) ---------------------------------------- (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 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) S tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2))) QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_2, c4_1, c8_2, c8_1, c1_2, c1_1, c2_2, c2_1 ---------------------------------------- (77) 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, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) We considered the (Usable) Rules:none And the Tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) 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(QSORT(x_1)) = [1] POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1)) = x_1 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)) = 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(nil) = [1] POL(s(x_1)) = 0 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (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 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) S tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2))) QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_2, c4_1, c8_2, c8_1, c1_2, c1_1, c2_2, c2_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) by FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) ---------------------------------------- (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 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) S tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2))) QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_2, c8_1, c1_2, c1_1, c2_2, c2_1, c4_2 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) by FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) ---------------------------------------- (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 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) S tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2))) QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_2, c1_1, c2_2, c2_1, c4_2, c8_2 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2))) by QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) ---------------------------------------- (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 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) S tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_2, c2_1, c4_2, c8_2, c1_2 ---------------------------------------- (85) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) ---------------------------------------- (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 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) S tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_2, c2_1, c4_2, c8_2, c1_2 ---------------------------------------- (87) 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, 0, 0, x1)), FILTERLOW(0, 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) if2(true, z0, z1, z2) -> filterhigh(z0, z2) if1(true, z0, z1, z2) -> filterlow(z0, z2) if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2)) And the Tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) 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(QSORT(x_1)) = [1] + x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_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_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(nil) = [1] POL(s(x_1)) = [1] POL(true) = 0 POL(ys) = 0 ---------------------------------------- (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 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) S tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_2, c2_1, c4_2, c8_2, c1_2 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) by QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) ---------------------------------------- (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 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) S tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2 ---------------------------------------- (91) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) ---------------------------------------- (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 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) S tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2 ---------------------------------------- (93) 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)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) 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(QSORT(x_1)) = [1] + x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1)) = x_1 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_1 + x_2 POL(ge(x_1, x_2)) = [1] POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = x_1 + x_4 POL(nil) = [1] POL(s(x_1)) = 0 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (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 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) S tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2 ---------------------------------------- (95) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) by QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) ---------------------------------------- (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 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) S tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2 ---------------------------------------- (97) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) by QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) ---------------------------------------- (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 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) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) S tuples: IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2 ---------------------------------------- (99) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) by IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) ---------------------------------------- (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 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) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) S tuples: IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1 ---------------------------------------- (101) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) by IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) ---------------------------------------- (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 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) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) S tuples: IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4 Compound Symbols: c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1 ---------------------------------------- (103) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) by IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) ---------------------------------------- (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 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) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) S tuples: IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c13(GE(z0, z1)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4 Compound Symbols: c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1 ---------------------------------------- (105) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) by IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) ---------------------------------------- (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 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)) -> c13(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) S tuples: GE(s(z0), s(z1)) -> c13(GE(z0, z1)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4 Compound Symbols: c13_1, c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1 ---------------------------------------- (107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace GE(s(z0), s(z1)) -> c13(GE(z0, z1)) by GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) ---------------------------------------- (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 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) -> c15(APPEND(z1, ys)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) S tuples: FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2 Compound Symbols: c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1 ---------------------------------------- (109) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 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 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) -> c15(APPEND(z1, ys)) FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) S tuples: FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) K tuples: APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2 Compound Symbols: c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1 ---------------------------------------- (111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) by APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) ---------------------------------------- (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 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(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) S tuples: FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) K tuples: QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2 Compound Symbols: c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1 ---------------------------------------- (113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) by QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) ---------------------------------------- (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 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(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) S tuples: FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) K tuples: QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2 Compound Symbols: c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1 ---------------------------------------- (115) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) by QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) ---------------------------------------- (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 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(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) S tuples: FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(GE(s(x0), s(x1))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2 Compound Symbols: c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1 ---------------------------------------- (117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1))) by FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) ---------------------------------------- (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 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(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) S tuples: FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(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)) -> c8(GE(s(x1), s(x0))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2 Compound Symbols: c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1 ---------------------------------------- (119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0))) by FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) ---------------------------------------- (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 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(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) S tuples: FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2 Compound Symbols: c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1 ---------------------------------------- (121) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) by QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) ---------------------------------------- (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 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(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) S tuples: FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2 Compound Symbols: c4_1, c8_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c1_1 ---------------------------------------- (123) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) by QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) ---------------------------------------- (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 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(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) S tuples: FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2 Compound Symbols: c4_1, c8_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c1_1, c2_1 ---------------------------------------- (125) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) by IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) ---------------------------------------- (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 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(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) S tuples: FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2 Compound Symbols: c4_1, c8_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c1_1, c2_1 ---------------------------------------- (127) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) by FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) ---------------------------------------- (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 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(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) S tuples: FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2 Compound Symbols: c4_1, c8_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c1_1, c2_1 ---------------------------------------- (129) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) by IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) ---------------------------------------- (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 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(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) S tuples: FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2 Compound Symbols: c4_1, c8_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c1_1, c2_1 ---------------------------------------- (131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) by IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) ---------------------------------------- (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 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(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) S tuples: FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2 Compound Symbols: c4_1, c8_1, c4_2, c8_2, c1_2, c2_2, c6_1, c9_1, c10_1, c13_1, c15_1, c1_1, c2_1, c5_1 ---------------------------------------- (133) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) by IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) ---------------------------------------- (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 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(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) S tuples: FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2 Compound Symbols: c4_1, c8_1, c4_2, c8_2, c1_2, c2_2, c6_1, c9_1, c10_1, c13_1, c15_1, c1_1, c2_1, c5_1 ---------------------------------------- (135) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) by FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) ---------------------------------------- (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 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: FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) S tuples: FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERHIGH_2, FILTERLOW_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2 Compound Symbols: c8_1, c4_2, c8_2, c1_2, c2_2, c6_1, c9_1, c10_1, c13_1, c4_1, c15_1, c1_1, c2_1, c5_1 ---------------------------------------- (137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) by IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) ---------------------------------------- (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 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: FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) S tuples: FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERHIGH_2, FILTERLOW_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2 Compound Symbols: c8_1, c4_2, c8_2, c1_2, c2_2, c6_1, c9_1, c10_1, c13_1, c4_1, c15_1, c1_1, c2_1, c5_1 ---------------------------------------- (139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) by IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) ---------------------------------------- (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 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: FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) S tuples: FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERHIGH_2, FILTERLOW_2, QSORT_1, IF2_4, GE_2, APPEND_2, IF1_4 Compound Symbols: c8_1, c4_2, c8_2, c1_2, c2_2, c9_1, c10_1, c13_1, c4_1, c15_1, c1_1, c2_1, c5_1, c6_1 ---------------------------------------- (141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) by IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) ---------------------------------------- (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 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: FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) S tuples: FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERHIGH_2, FILTERLOW_2, QSORT_1, IF2_4, GE_2, APPEND_2, IF1_4 Compound Symbols: c8_1, c4_2, c8_2, c1_2, c2_2, c9_1, c10_1, c13_1, c4_1, c15_1, c1_1, c2_1, c5_1, c6_1 ---------------------------------------- (143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) by FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(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 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: FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) S tuples: FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERHIGH_2, FILTERLOW_2, QSORT_1, IF2_4, GE_2, APPEND_2, IF1_4 Compound Symbols: c8_1, c4_2, c8_2, c1_2, c2_2, c9_1, c10_1, c13_1, c4_1, c15_1, c1_1, c2_1, c5_1, c6_1 ---------------------------------------- (145) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) by IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(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 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: FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) S tuples: FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERHIGH_2, FILTERLOW_2, QSORT_1, IF2_4, GE_2, APPEND_2, IF1_4 Compound Symbols: c8_1, c4_2, c8_2, c1_2, c2_2, c9_1, c10_1, c13_1, c4_1, c15_1, c1_1, c2_1, c5_1, c6_1 ---------------------------------------- (147) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) by IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) ---------------------------------------- (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 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: FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) S tuples: FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERHIGH_2, FILTERLOW_2, QSORT_1, IF2_4, GE_2, APPEND_2, IF1_4 Compound Symbols: c8_1, c4_2, c8_2, c1_2, c2_2, c10_1, c13_1, c4_1, c15_1, c1_1, c2_1, c5_1, c6_1, c9_1 ---------------------------------------- (149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) by IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), 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 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: FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) S tuples: FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERHIGH_2, FILTERLOW_2, QSORT_1, IF2_4, GE_2, APPEND_2, IF1_4 Compound Symbols: c8_1, c4_2, c8_2, c1_2, c2_2, c10_1, c13_1, c4_1, c15_1, c1_1, c2_1, c5_1, c6_1, c9_1 ---------------------------------------- (151) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) by FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) ---------------------------------------- (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 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(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) S tuples: FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF2_4, GE_2, APPEND_2, IF1_4 Compound Symbols: c4_2, c8_2, c1_2, c2_2, c10_1, c13_1, c4_1, c8_1, c15_1, c1_1, c2_1, c5_1, c6_1, c9_1 ---------------------------------------- (153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) by IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) ---------------------------------------- (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 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(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) S tuples: FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF2_4, GE_2, APPEND_2, IF1_4 Compound Symbols: c4_2, c8_2, c1_2, c2_2, c10_1, c13_1, c4_1, c8_1, c15_1, c1_1, c2_1, c5_1, c6_1, c9_1 ---------------------------------------- (155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) by IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) ---------------------------------------- (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 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(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) S tuples: FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, FILTERHIGH_2, QSORT_1, GE_2, APPEND_2, IF1_4, IF2_4 Compound Symbols: c4_2, c8_2, c1_2, c2_2, c13_1, c4_1, c8_1, c15_1, c1_1, c2_1, c5_1, c6_1, c9_1, c10_1 ---------------------------------------- (157) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) by GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) ---------------------------------------- (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 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(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) S tuples: FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERLOW_2, FILTERHIGH_2, QSORT_1, APPEND_2, IF1_4, IF2_4, GE_2 Compound Symbols: c4_2, c8_2, c1_2, c2_2, c4_1, c8_1, c15_1, c1_1, c2_1, c5_1, c6_1, c9_1, c10_1, c13_1 ---------------------------------------- (159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) by FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1))))) ---------------------------------------- (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 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: FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1))))) S tuples: FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1))))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: FILTERHIGH_2, QSORT_1, FILTERLOW_2, APPEND_2, IF1_4, IF2_4, GE_2 Compound Symbols: c8_2, c1_2, c2_2, c4_1, c8_1, c15_1, c1_1, c2_1, c5_1, c6_1, c9_1, c10_1, c13_1, c4_2 ---------------------------------------- (161) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) by FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1))))) ---------------------------------------- (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 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: QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1))))) S tuples: QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1))))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: QSORT_1, FILTERLOW_2, FILTERHIGH_2, APPEND_2, IF1_4, IF2_4, GE_2 Compound Symbols: c1_2, c2_2, c4_1, c8_1, c15_1, c1_1, c2_1, c5_1, c6_1, c9_1, c10_1, c13_1, c4_2, c8_2 ---------------------------------------- (163) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) by FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2))) FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1))) FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1)))) FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1)))) ---------------------------------------- (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 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: QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2))) FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1))) FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1)))) FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1)))) S tuples: QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2))) FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1))) FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1)))) FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1)))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: QSORT_1, FILTERLOW_2, FILTERHIGH_2, APPEND_2, IF1_4, IF2_4, GE_2 Compound Symbols: c1_2, c2_2, c4_1, c8_1, c15_1, c1_1, c2_1, c5_1, c6_1, c9_1, c10_1, c13_1, c4_2, c8_2 ---------------------------------------- (165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) by FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) ---------------------------------------- (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 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: QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2))) FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1))) FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1)))) FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) S tuples: QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2))) FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1))) FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1)))) FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: QSORT_1, FILTERHIGH_2, APPEND_2, FILTERLOW_2, IF1_4, IF2_4, GE_2 Compound Symbols: c1_2, c2_2, c8_1, c15_1, c1_1, c2_1, c4_1, c5_1, c6_1, c9_1, c10_1, c13_1, c4_2, c8_2 ---------------------------------------- (167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) by FILTERHIGH(s(0), cons(s(z0), cons(0, y1))) -> c8(IF2(true, s(0), s(z0), cons(0, y1))) FILTERHIGH(s(0), cons(s(z0), cons(s(y1), y2))) -> c8(IF2(true, s(0), s(z0), cons(s(y1), y2))) ---------------------------------------- (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 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: QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2))) FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1))) FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1)))) FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) FILTERHIGH(s(0), cons(s(z0), cons(0, y1))) -> c8(IF2(true, s(0), s(z0), cons(0, y1))) FILTERHIGH(s(0), cons(s(z0), cons(s(y1), y2))) -> c8(IF2(true, s(0), s(z0), cons(s(y1), y2))) S tuples: QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2))) FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1))) FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1)))) FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) FILTERHIGH(s(0), cons(s(z0), cons(0, y1))) -> c8(IF2(true, s(0), s(z0), cons(0, y1))) FILTERHIGH(s(0), cons(s(z0), cons(s(y1), y2))) -> c8(IF2(true, s(0), s(z0), cons(s(y1), y2))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: QSORT_1, FILTERHIGH_2, APPEND_2, FILTERLOW_2, IF1_4, IF2_4, GE_2 Compound Symbols: c1_2, c2_2, c8_1, c15_1, c1_1, c2_1, c4_1, c5_1, c6_1, c9_1, c10_1, c13_1, c4_2, c8_2 ---------------------------------------- (169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) by FILTERHIGH(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) FILTERHIGH(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(0), y1))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) ---------------------------------------- (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 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: QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2))) FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1))) FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1)))) FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) FILTERHIGH(s(0), cons(s(z0), cons(0, y1))) -> c8(IF2(true, s(0), s(z0), cons(0, y1))) FILTERHIGH(s(0), cons(s(z0), cons(s(y1), y2))) -> c8(IF2(true, s(0), s(z0), cons(s(y1), y2))) FILTERHIGH(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) FILTERHIGH(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(0), y1))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) S tuples: QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2))) FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1))) FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1)))) FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) FILTERHIGH(s(0), cons(s(z0), cons(0, y1))) -> c8(IF2(true, s(0), s(z0), cons(0, y1))) FILTERHIGH(s(0), cons(s(z0), cons(s(y1), y2))) -> c8(IF2(true, s(0), s(z0), cons(s(y1), y2))) FILTERHIGH(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) FILTERHIGH(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(0), y1))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: QSORT_1, APPEND_2, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2 Compound Symbols: c1_2, c2_2, c15_1, c1_1, c2_1, c4_1, c8_1, c5_1, c6_1, c9_1, c10_1, c13_1, c4_2, c8_2 ---------------------------------------- (171) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) by APPEND(cons(z0, cons(z1, cons(y1, y2))), ys) -> c15(APPEND(cons(z1, cons(y1, y2)), ys)) ---------------------------------------- (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 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: QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2))) FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1))) FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1)))) FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) FILTERHIGH(s(0), cons(s(z0), cons(0, y1))) -> c8(IF2(true, s(0), s(z0), cons(0, y1))) FILTERHIGH(s(0), cons(s(z0), cons(s(y1), y2))) -> c8(IF2(true, s(0), s(z0), cons(s(y1), y2))) FILTERHIGH(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) FILTERHIGH(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(0), y1))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) APPEND(cons(z0, cons(z1, cons(y1, y2))), ys) -> c15(APPEND(cons(z1, cons(y1, y2)), ys)) S tuples: QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2))) FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1))) FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1)))) FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) FILTERHIGH(s(0), cons(s(z0), cons(0, y1))) -> c8(IF2(true, s(0), s(z0), cons(0, y1))) FILTERHIGH(s(0), cons(s(z0), cons(s(y1), y2))) -> c8(IF2(true, s(0), s(z0), cons(s(y1), y2))) FILTERHIGH(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) FILTERHIGH(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(0), y1))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) APPEND(cons(z0, cons(z1, cons(y1, y2))), ys) -> c15(APPEND(cons(z1, cons(y1, y2)), ys)) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2, APPEND_2 Compound Symbols: c1_2, c2_2, c1_1, c2_1, c4_1, c8_1, c5_1, c6_1, c9_1, c10_1, c13_1, c4_2, c8_2, c15_1 ---------------------------------------- (173) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) by QSORT(cons(0, cons(0, y1))) -> c1(FILTERLOW(0, cons(0, cons(0, y1)))) QSORT(cons(0, cons(s(y0), y1))) -> c1(FILTERLOW(0, cons(0, cons(s(y0), y1)))) ---------------------------------------- (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 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: QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0))) QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0))) QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2))) FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1))) FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1)))) FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) FILTERHIGH(s(0), cons(s(z0), cons(0, y1))) -> c8(IF2(true, s(0), s(z0), cons(0, y1))) FILTERHIGH(s(0), cons(s(z0), cons(s(y1), y2))) -> c8(IF2(true, s(0), s(z0), cons(s(y1), y2))) FILTERHIGH(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) FILTERHIGH(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(0), y1))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) APPEND(cons(z0, cons(z1, cons(y1, y2))), ys) -> c15(APPEND(cons(z1, cons(y1, y2)), ys)) QSORT(cons(0, cons(0, y1))) -> c1(FILTERLOW(0, cons(0, cons(0, y1)))) QSORT(cons(0, cons(s(y0), y1))) -> c1(FILTERLOW(0, cons(0, cons(s(y0), y1)))) S tuples: QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1))) QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1))) FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1)))) FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1))) IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1))) FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1))) FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1))) FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1)))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1)))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1)))) IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1)))) FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2))) FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1))) IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1))) FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2))) IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1))) IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1))) IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1))) FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1))) FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2))) IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1)))) FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0)))) FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1))))) FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2))) FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1))) FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1)))) FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1))) FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) FILTERHIGH(s(0), cons(s(z0), cons(0, y1))) -> c8(IF2(true, s(0), s(z0), cons(0, y1))) FILTERHIGH(s(0), cons(s(z0), cons(s(y1), y2))) -> c8(IF2(true, s(0), s(z0), cons(s(y1), y2))) FILTERHIGH(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) FILTERHIGH(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(0), y1))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) K tuples: QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1))) QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1))) QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1))) QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1))) QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1))) APPEND(cons(z0, cons(z1, cons(y1, y2))), ys) -> c15(APPEND(cons(z1, cons(y1, y2)), ys)) QSORT(cons(0, cons(0, y1))) -> c1(FILTERLOW(0, cons(0, cons(0, y1)))) QSORT(cons(0, cons(s(y0), y1))) -> c1(FILTERLOW(0, cons(0, cons(s(y0), y1)))) Defined Rule Symbols: ge_2, filterlow_2, if1_4, filterhigh_2, if2_4 Defined Pair Symbols: QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2, APPEND_2 Compound Symbols: c1_2, c2_2, c1_1, c2_1, c4_1, c8_1, c5_1, c6_1, c9_1, c10_1, c13_1, c4_2, c8_2, c15_1